![INTRODUCTION
In this introduction, we summarise the mathematical career of Noel Baker
and indicate how the papers in this volume relate to his work. Much of
the material is taken from the obituary of Noel Baker that appeared in the
Bulletin of the London Mathematical Society [17].
Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20
May 2001. He grew up in Australia and was first introduced to the theory
of iteration by his MSc supervisor, George Szekeres, who suggested that he
work on the functional equation
f(f(z)) = F(z),
where f and F are analytic functions. In his first mathematical paper (1),
Noel used the theory of iteration of analytic functions, which had been devel-
oped principally by Fatou and Julia and which was not well known at that
time. He used this theory to show, amongst other things, that if F belongs
to a certain class of entire functions, which includes the exponential function,
then the above equation has no entire solution. This first paper also contains
examples that were constructed using Wiman-Valiron theory. Throughout his
career Noel was to find ever more techniques from classical complex analysis
that can usefully be applied to iteration theory.
In 1955 Noel won a German government scholarship to the University of
Tübingen, where he worked under Hellmuth Kneser. Noel’s doctoral thesis,
published in (2), continued his study of functional equations. From 1957 to
1959, Noel taught mathematics at the University of Alberta in Edmonton,
Canada. In 1959 he moved to Imperial College London, where he remained
until retirement in 1997.
In his research, Noel worked on many problems in complex analysis and had
a wide range of collaborators, but iteration theory, his great love, was for
many years a lone interest. However, when the subject was reborn around
1980, partly as a result of the advent of accessible computer graphics, it
became clear to the new adherents that Noel had for many years been quietly
and carefully completing the foundations begun earlier in the century by the
French mathematicians Pierre Fatou and Gaston Julia. He had also pointed
the way towards many future developments, both by proving new results and
by posing challenging problems. In the explosion of research on iteration
theory that took place in the subsequent years, many of the papers published
on iteration made reference to Noel’s work and he received many invitations
to speak at international conferences on iteration. At these he would often
appear reserved, much preferring to let others speak about the latest work,
ix](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-13-320.jpg)
![x INTRODUCTION
even though he was the acknowledged authority on many matters, and the
person whose judgement about the validity of a new proof was always sought.
Noel continued his research after his retirement and one of his last papers was
dedicated to George Szekeres on the occasion of the latter’s 90th birthday.
Noel’s early work on functional equations led him to consider problems about
periodic points, which play a very important role in complex dynamics. It
was already known that for an entire function there must be infinitely many
periodic points of period p, for all p ≥ 2, but Noel considered the unsolved
problem of the existence of periodic points of a given exact period. He showed
in (6) that for all non-linear entire functions there exist periodic points of
exact period p, for all p with at most one exception; for example, f(z) = z+ez
has no fixed points. In a later paper (13) Noel showed that for a polynomial
the only possible exceptional value in this result is p = 2, the corresponding
exceptional functions being f(z) = z2
− z and other quadratics ‘similar’ to
this one. He also conjectured that for a transcendental entire function the
only possible exceptional value is p = 1, and this was proved by Bergweiler [2].
We now describe the origins of complex dynamics. Let f be a rational function
of degree at least 2 or a transcendental entire function. The set of points near
which the sequence of iterates fn
forms a normal family is called the Fatou set
F(f) and its complement is called the Julia set J(f). Roughly speaking, the
dynamics are stable on the Fatou set and chaotic on the Julia set. Also, the
Julia set often exhibits great topological complexity as well as ‘self-similarity’;
for example, the paper in this volume by Devaney et al discusses a family of
rational functions whose Julia sets in some cases contain Cantor sets of curves
and in other cases contain Sierpinski curves.
The fundamental properties of the sets F(f) and J(f) were first established
for rational functions in [13] and [8], and for transcendental entire functions
in [9]. In the last paper, Fatou studied the iteration of transcendental entire
functions in some detail, giving examples that pointed to significant differ-
ences to the theory that had been developed for rational functions. He asked
the following fundamental questions about a transcendental entire function f:
1. Are the repelling periodic points of f dense in J(f)?
2. Are there examples where J(f) = C? In particular, is this true for
f(z) = ez
?
3. Can J(f) be totally disconnected?
4. Must J(f) contain infinitely many unbounded analytic curves, at each
point of which fn
→ ∞?
Question 1 is of great theoretical importance, and it had been answered ‘yes’
for rational functions by both Fatou and Julia. Fatou had also given an
example of a rational function f for which J(f) is totally disconnected, and
Lattès [14] an example for which J(f) = C. Most of Fatou’s questions were
solved by Noel during the decade 1965–1975, as we now indicate.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-14-320.jpg)
![INTRODUCTION xi
The first question was answered in the affirmative in the paper (22), which is
of fundamental importance in complex dynamics and appropriately dedicated
to Hellmuth Kneser. Here, Noel called on a deep covering theorem due to
Ahlfors (see [11, page 148]) to show that arbitrarily close to each point of J(f)
there is a repelling periodic point of f. From this, he deduced the general
result that if f is any non-linear entire function, then the set of entire functions
that commute with f is countable. Many authors have tried to simplify the
proof in (22) that the repelling periodic points are dense in J(f), in order to
avoid the deep theorem of Ahlfors. Eventually, more elementary proofs based
on a renormalisation technique were given by Schwick [18], Bargmann [1],
and Berteloot and Duval [4].
Two years later, in (25), Noel answered the first part of Fatou’s second ques-
tion by showing that there is a function of the form f(z) = kzez
, where k 0,
such that J(f) = C. A proof that if f(z) = ez
, then we have J(f) = C was
given ten years later by Misiurewicz [15].
Noel answered Fatou’s third question in the negative in (32). If J(f) is totally
disconnected, then F(f) must have a single unbounded multiply connected
component. Noel had already constructed in (9) an example of a transcen-
dental entire function for which F(f) has at least one multiply connected
component. This function was of the form
f(z) = Cz2
∞
Y
n=1
1 +
z
rn
,
in which the positive constants r1 r2 . . . have the property that
f(An) ⊂ An+1, where An = {z : r2
n |z| r
1/2
n+1}.
However, Noel did not determine in (9) whether F(f) has a single unbounded
multiply connected component or a sequence of bounded multiply connected
components. In (33) he used Schottky’s theorem [11, page 169], yet another
result from classical complex analysis, to show that the latter must be the
case. This solved another important problem in complex dynamics, open
since the work of Fatou and Julia, by showing that the above function has
a sequence of wandering domains, that is, distinct components Un of F(f)
such that f(Un) ⊂ Un+1, for n = 1, 2, . . . . In contrast, Sullivan [16] showed
that rational functions do not have wandering domains. The paper (32),
written later than but published earlier than (33), used Schottky’s theorem
once again to show that a transcendental entire function cannot have an
unbounded multiply connected component of F(f), thus proving that J(f)
can never be totally disconnected.
The results in (32) and (33) led to much further work. In (53), Noel showed
that wandering domains for transcendental entire functions may be infinitely
connected. For many years it was not known whether such wandering domains](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-15-320.jpg)
![xii INTRODUCTION
could be finitely connected. In this volume, Kisaka and Shishikura show that
they can in fact have any given finite connectivity.
The result in (32) shows that if f is a transcendental entire function, then
J(f) must contain a continuum, so its Hausdorff dimension dimHJ(f) is at
least 1. It remains an open question whether dimHJ(f) = 1 is possible. In
this volume, there is a survey article on dimensions of Julia sets by Stallard,
complemented by a survey article on fractal measures and ergodic theory by
Kotus and Urbański.
Noel’s wandering domains example mentioned earlier shows that the answer
to Fatou’s fourth question (as stated here) is ‘no’. However, the structure of
the ‘escaping set’, where fn
→ ∞, continues to stimulate much work, includ-
ing the paper by Rottenfusser and Schleicher that appears in this volume.
Sullivan’s remarkable result [16] that rational functions do not have wan-
dering domains was proved using new techniques based on quasiconformal
conjugacy. Noel quickly saw that these new techniques would also apply to
various families of transcendental entire functions, and a proof that exponen-
tial functions have no wandering domains appeared in (49). This was one of a
number of papers at that time that established many of the basic dynamical
properties of the exponential family and began the description of the corre-
sponding parameter space, the ‘exponential Mandelbrot set’, which has since
been the subject of much study — see, for example, the paper by Rempe and
Schleicher in this volume.
In (41), Noel initiated another major development by showing that if a tran-
scendental entire function f has order of growth at most 1/2, minimal type,
then F(f) has no unbounded invariant components, and he also gave a more
restrictive condition on the maximum modulus of f that forces every compo-
nent of F(f) to be bounded. The question of whether the latter conclusion
follows from order at most 1/2, minimal type, remains open, though many
authors have obtained partial results in this direction; this volume contains
a survey article on this problem by Hinkkanen.
A key step in Noel’s proof in (41) is to exclude unbounded invariant com-
ponents of F(f) in which fn
→ ∞. He did this by establishing estimates
for the growth of iterates in such components, which he later refined in (57).
In recognition of his work on Fatou components of this type, Eremenko and
Lyubich introduced the name Baker domain for such components in [7]. In
this fundamental paper, Eremenko and Lyubich showed that if the set S(f) of
inverse function singularities of a transcendental entire function f is bounded,
then f has no Baker domains and if S(f) is finite, then f has no wandering
domains; see also [10]. A survey article on Baker domains by Rippon appears
in this volume.
Yet another fundamental contribution to the iteration of transcendental entire
functions came in the papers (65), (73) and (74). Once again an unbounded](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-16-320.jpg)
![INTRODUCTION xiii
invariant component U of F(f) was considered, but now the aim was to
describe the nature of the boundary of U. Some special cases had been
investigated by other authors, following the appearance of computer pictures
of Julia sets, but Noel and his students Weinreich and Domı́nguez attacked
the general case. In (65), it was shown that
• if U is not a Baker domain (that is, U is an attracting basin, a para-
bolic basin, or a Siegel disc), then ∂U is sufficiently complicated that
∞ belongs to the impression of every prime end of U;
• if ∂U is a Jordan curve in the extended complex plane Ĉ (and such
U do exist), then not only must U be a Baker domain, but f must be
univalent in U.
The key tool introduced in this work arises from the fact that if Ψ is a
conformal map from the unit disc D onto U, then Ψ−1
◦ f ◦ Ψ is an inner
function, that is, an analytic self-map of D whose angular limits have modulus
1 almost everywhere on ∂D. The paper (65) initiated a version of Fatou-Julia
theory for inner functions, a topic now of interest in its own right, and this
theory was taken further in (73). Further results on this theory are given in
the paper by Bargmann in this volume.
Many of Noel’s final papers are joint papers with his last student, Domı́nguez,
and concern the connectedness properties of the Julia set. Many of these
results are described and extended in the paper by Domı́nguez and Fagella
in this volume.
Fatou-Julia theory of the iteration of general transcendental meromorphic
functions was established in the fundamental papers (62), (63), (64) and (66)
by Baker, Kotus and Lü. The Fatou set F(f) is here taken to be the set
of points near which the iterates fn
are defined and form a normal family,
and then J(f) = Ĉ F(f). Many of the basic results turn out to be similar
to those for rational and entire functions, but there are some striking differ-
ences. For example, in (62) the authors showed that J(f) is once again the
closure of the repelling periodic points of f, and this fact is used to give a
complete classification of those transcendental meromorphic functions, such
as f(z) = tan z, for which J(f) is a subset of the real line; there are no tran-
scendental entire functions for which the Julia set is contained in the real line.
Then, in (63), they used techniques from approximation theory, pioneered by
Eremenko and Lyubich [6], to construct transcendental meromorphic func-
tions with wandering domains of all possible connectivities.
The question of periodic components was taken up in (64), where the authors
showed that precisely five possible types can arise for a transcendental mero-
morphic function, namely, attracting basins, parabolic basins, Siegel discs,
Herman rings and Baker domains. Moreover, any invariant components of
F(f) must be simply connected, doubly connected, or infinitely connected.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-17-320.jpg)
![xiv INTRODUCTION
But perhaps the most striking result here was the construction of a tran-
scendental meromorphic function f with a preperiodic component of F(f)
of any given finite connectivity. This construction used the powerful tech-
nique of quasiconformal surgery, introduced by Shishikura [19], which also
appears in many of the papers in this volume — namely, those by Drasin
and Langley, Domı́nguez and Fagella, and Kisaka and Shishikura. Finally,
in (66), Sullivan’s method of quasiconformal conjugacy was adapted to show
that a transcendental meromorphic function of finite type has no wandering
domains. These four papers opened a new and fruitful area of research, made
even more accessible by the excellent survey article [3], which appeared soon
after.
One of the differences between the iteration of entire functions and mero-
morphic functions is the number of completely invariant components of the
Fatou set that can occur. In (24) Noel proved that a transcendental entire
function can have at most one completely invariant component of the Fatou
set. In (64) Baker, Kotus and Lü proved that a transcendental meromorphic
function of finite type can have at most two completely invariant Fatou com-
ponents, and in this volume it is shown by Bergweiler and Eremenko that,
in these circumstances, the Julia set must be a Jordan curve. (An example
of a function with these properties is f(z) = tan z.) It is an open question
whether a general transcendental meromorphic function can have at most two
completely invariant Fatou components.
Fatou-Julia theory can be developed in many further directions. For a tran-
scendental meromorphic function f, the iterates fn
need not be meromorphic.
It is desirable, however, to have a closed system of iterates, so that we can
consider, for example, the Fatou set of fn
, for n ≥ 2. To obtain such a
system, Noel’s student Herring [12], and independently Bolsch [5], developed
Fatou-Julia theory for functions, such as f(z) = etan z
, which are meromor-
phic outside certain compact totally disconnected subsets of Ĉ. Much of
this theory, and its subsequent developments, is expounded in Noel’s last
papers (75), (77), (78) and (79).
This volume also contains papers that, while not explicitly about complex
dynamics, are on closely related topics. The paper by Hayman and Hinkka-
nen is concerned with the growth of meromorphic functions that belong to
certain normal families, the paper by Beardon and Minda classifies conformal
automorphisms of finitely connected regions of the plane, and the paper by
Szekeres is on possible connections between ‘regular growth’ and Abel’s func-
tional equation, a topic in which Noel had a great interest. Finally, the paper
by Bullett and Freiburger is on the theory of holomorphic correspondences, a
generalisation of complex dynamics. Here they investigate, for the first time,
holomorphic correspondences that involve transcendental entire functions.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-18-320.jpg)
![INTRODUCTION xix
(78) Limit functions in wandering domains of meromorphic functions, Ann.
Acad. Sci. Fenn. Math. 27 (2002) 499–505.
(79) (with P. DOMÍNGUEZ and M. HERRING) Functions meromorphic
outside a small set: completely invariant domains, Complex Variables
Theory Appl. 49 (2004) 95–100.
References
[1] D. Bargmann, Simple proofs of some fundamental properties of the Julia set, Ergodic
Theory Dynam. Systems 19 (1999) 553–558.
[2] W. Bergweiler, Periodic points of entire functions: proof of a conjecture of Baker,
Complex Variables Theory Appl. 17 (1991) 57–72.
[3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993)
151–188.
[4] F. Berteloot and J. Duval, Une démonstration directe de la densité de cycles répulsif
dans l’ensemble de Julia, Complex analysis and geometry (Paris 1997) , Progr. Math.
188 (Birkhäuser, Basel, 2000) 221–222.
[5] A. Bolsch, Repulsive periodic points of meromorphic functions, Complex Variables
Theory Appl. 31 (1996) 75–79.
[6] A. E. Eremenko and M. Yu. Lyubich, Examples of entire functions with pathological
dynamics, J. London Math. Soc. 36 (1987) 458–468.
[7] A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire
functions, Ann. Inst. Fourier (Grenoble) 42 (1992) 989–1020.
[8] P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France 47 (1919) 161–271;
48 (1920) 33–94, 208–314.
[9] P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Math. 47 (1926)
337–370.
[10] L. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions,
Ergodic Theory Dynam. Systems 6 (1986) 183–192.
[11] W. K. Hayman, Meromorphic functions (Clarendon Press, Oxford, 1964).
[12] M. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Univer-
sity of London (1994).
[13] G. Julia, Mémoire sur l’itération des fonctions rationelles, J. Math. Pure Appl. 1
(1918) 47–245.
[14] S. Lattès, Sur l’itération des substitutions rationelles et les fonctions de Poincaré, C.R.
Acad. Sci. Paris Sér. I Math. 166 (1918) 26–28 (Errata: p. 88).
[15] M. Misiurewicz, On iterates of ez
, Ergodic Theory Dynam. Systems 1 (1981) 103–106.
[16] D. Sullivan, Quasiconformal homeomorphisms and dynamics I, Ann. Math. 122 (2)
(1985) 401–418.
[17] P. J. Rippon, Obituary: Irvine Noel Baker 1932–2001, Bull. London Math. Soc. 37
(2005) 301–315.
[18] W. Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc. 29
(1997) 314–316.
[19] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Scient. Ec.
Norm. Sup. (4) 20 (1987) 1–29.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-23-320.jpg)
![ITERATION OF INNER FUNCTIONS AND BOUNDARIES
OF COMPONENTS OF THE FATOU SET
DETLEF BARGMANN
Abstract. Let D be an unbounded invariant component of the Fatou
set of a transcendental entire function f. Let φ : D → D be a Riemann
map. Then the set Θ := {θ ∈ ∂D : limr→1 φ(rθ) = ∞} is closely related
to the Julia set of the corresponding inner function g := φ−1
◦ f ◦ φ. In
the first part of the paper we further develop the theory of Julia sets of
inner functions and the dynamical behaviour on their Fatou sets. In the
second part we apply these results to iteration of entire functions by using
the above relation and obtain some new results about the boundaries of
components of the Fatou set of an entire function.
1. Introduction
Dynamics of inner functions have turned out to be a very useful tool to
study the boundary structure of unbounded invariant components of the Fa-
tou set of a transcendental entire function. The key method in this area has
been developed by I.N. Baker and P. Domı́nguez [3]. We recall some of their
techniques and results in Subsection 1.1 and then state the aims and results
of this paper in Subsection 1.2
1.1. The method of Baker-Domı́nguez. Let f be a transcendental entire
function, with Fatou set F(f) and Julia set J (f); see [7] for background
information on these concepts. Suppose that D is an unbounded invariant
component of F(f). Then D is simply connected [2, Theorem 1], which
implies that there exists a biholomorphic (Riemann) map φ : D → D. Then
g := φ−1
◦ f ◦ φ
is an inner function, i.e. a holomorphic self-map of the unit disk such that
lim
r→1
|g(r exp(2πiα))| = 1,
for almost every α ∈ [0, 1]. According to Fatou’s theorem [18, p. 139] this
implies that limr→1 g(r exp(2πiα)) exists and is contained in ∂D, for almost
every α ∈ [0, 1].
Keywords: inner function, Julia set, Fatou set, boundary, radial limit, Baker domain.
AMS subject classification: 30D05, 58F08.
1](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-25-320.jpg)
![2 DETLEF BARGMANN
If f|D is a proper self-map of D, then g is a (finite) Blaschke product, i.e.
there exist m, n ∈ N0, λ ∈ ∂D, and a1, . . . , an ∈ D {0} such that
g(z) = λzm
n
Y
j=1
z − aj
1 − ajz
,
for each z ∈ D.
Since D is unbounded f|D need not be a proper self-map of D. In this
case g has at least one singularity on the boundary of the unit disk.
Definition 1.1. Let g be an inner function of D. A point ζ ∈ ∂D is called a
singularity of g if g cannot be continued analytically to a neighbourhood of ζ.
Denote the set of all singularities of g by sing(g).
Throughout this paper we assume that an inner function is always contin-
ued to ĈD by the reflection principle, where Ĉ denotes the complex sphere,
and to ∂D sing(g) by analytic continuation.
It follows from the theory of inner functions that the composition of two
inner functions is again an inner function; see [3, Lemma 4]. In particular,
the n-th iterate gn
of an inner function g is an inner function. Now, the Julia
set of an inner function can be defined in the following way.
Definition 1.2. Let g be an inner function of the unit disk D. The Fa-
tou set F(g) of g is the set of all points z ∈ Ĉ for which there is an open
neighbourhood U ⊂ Ĉ of z such that U ∩ sing(gn
) = ∅, for each n ∈ N, and
{gn
|U : n ∈ N} is normal. The Julia set J (g) of g is the complement of F(g)
in Ĉ.
Remark 1.3. It follows from Montel’s theorem that J (g) ⊂ ∂D. Moreover,
for the case of a finite Blaschke product this definition coincides with the usual
definition of the Julia set of a rational function.
Baker and Domı́nguez initiated the study of Julia sets of inner functions
by proving the following result [3, Lemma 8].
Theorem 1.4 (Baker-Domı́nguez). Let g be an inner function of the unit
disk D. Then the following properties hold.
(1) g(F(g)) ⊂ F(g).
(2) If g is non-Möbius, then J (g) is a perfect set.
(3) If g is non-rational, then J (g) =
S
n∈N sing(gn).
The main tool in the proof of Theorem 1.4 is the the following lemma on
inner functions [3, Lemma 5], which we use later. For the definition of a Stolz
angle we refer the reader to [18, p. 6].
Lemma 1.5 (Baker-Domı́nguez). Let g be an inner function of the unit disk.
Suppose that ζ ∈ sing(g). Then, for each θ ∈ ∂D and each neighbourhood U of
ζ, there exist η ∈ U {ζ} and a path γ : [0, 1) → D such that limt→1 γ(t) = η
and g(γ(t)) → θ in a Stolz angle as t → 1.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-26-320.jpg)
![ITERATION OF INNER FUNCTIONS 3
Following Baker and Domı́nguez [3], we relate the boundary behaviour of
the Riemann map φ to the dynamical behaviour of the corresponding inner
function g. This process involves the sets
Ξ := {ζ ∈ ∂D : ∞ ∈ C(φ; ζ)},
and
Θ := {θ ∈ ∂D : lim
r→1
φ(rθ) = ∞}.
Here C(φ; ζ) denotes the cluster set of φ at ζ, i.e. the set of all values w ∈ Ĉ
for which there is sequence (zn)n∈N in D such that zn → ζ and φ(zn) → w as
n → ∞. Since D is unbounded the set Ξ is always non-empty. In general, it
is not known whether the set Θ is always non-empty. However, for the case
when D is a Baker domain, i.e. fn
|D → ∞ locally uniformly, it is easy to see
that Θ 6= ∅. Throughout this paper the sets Ξ and Θ will relate to a Riemann
mapping φ : D → D of the invariant Fatou component D of the function f
under consideration.
There is a close connection between Ξ and J (g).
Lemma 1.6. If f|D is not an automorphism of D, then J (g) ⊂ Ξ.
Proof. First, it is easy to see that Ξ is closed.
Case 1. Suppose that g is rational. Then g is a finite Blaschke product
and it is easy to see that Ξ is backward invariant under g. Since g is locally
injective on ∂D (see Remark 2.19) we conclude that Ξ is an infinite set. Hence
Ξ is a closed, backward invariant set which contains at least three points. This
implies that Ξ is a superset of J (g).
Case 2. Suppose that g is not rational. By Theorem 1.4 we need only show
that the singularities of the iterates of g are contained in Ξ. Let n ∈ N and
ζ be a singularity of gn
. Then C(gn
; ζ) = D (see for instance [13, Theorem
5.4]), which implies that
∞ ∈ D ⊂ C(φ ◦ gn
; ζ) = C(fn
◦ φ; ζ).
Thus we conclude that ∞ ∈ C(φ; ζ). 2
Using Lemma 1.5, Baker and Domı́nguez obtained a similar result for the
set Θ; see [3, Lemma 13].
Lemma 1.7 (Baker-Domı́nguez). Suppose that f|D is not an automorphism
of D and Θ 6= ∅. Then J (g) ⊂ Θ.
Hence the Julia set of the corresponding inner function g is a lower bound
for the size of the set Ξ and, provided that Θ 6= ∅, for the size of the set Θ.
This provides a strategy to show that the sets Ξ and Θ are equal to the unit
circle by showing that the Julia sets of the corresponding inner functions are
the unit circle. Baker and Domı́nguez gave two cases when J (g) = ∂D; see
[3, Lemma 9 and Lemma 10].](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-27-320.jpg)
![4 DETLEF BARGMANN
Theorem 1.8 (Baker-Domı́nguez). Let g be a non-Möbius inner function of
D with a fixed point p ∈ D. Then J (g) = ∂D.
Theorem 1.9 (Baker-Domı́nguez). Let g be an inner function of D. Suppose
that there exists p ∈ ∂D such that, for each z ∈ D, gn
(z) → p in an arbitrarily
small Stolz angle as n → ∞. Then g is non-Möbius and J (g) = ∂D.
The assumptions of Theorem 1.8 are satisfied for the corresponding inner
function g when D is an attracting domain of f. Theorem 1.9 can be applied
to the corresponding inner function g when D is a parabolic domain of f.
Taken together, these lead to the following result; see [4, Theorem 1] and [3,
Theorem 1.1].
Theorem 1.10 (Baker-Weinreich, Baker-Domı́nguez). Suppose that D is an
attracting domain, a parabolic domain, or a Siegel disk of f. Then
Ξ = ∂D and Θ 6= ∅ ⇒ Θ = ∂D.
This result does not carry over in general to Baker domains, since there
are examples of Baker domains (due to Baker-Weinreich and Bergweiler, see
Examples 3.4 and 3.5 in Subsection 3.1) whose boundaries are Jordan curves.
On the other hand, Baker and Weinreich proved [4, Theorem 4] that if D is a
Baker domain whose boundary is a Jordan curve, then f|D has to be univalent.
Thus it is natural to ask whether Theorem 1.10 holds for those Baker domains
for which f|D is not univalent. In this case, Baker and Domı́nguez proved
that Θ contains a perfect set and hence is infinite; see [3, Theorem 1.2].
1.2. Aims and results of this paper. The aim of this paper is twofold.
In Section 2 we further develop the theory of Julia sets of inner functions,
independently of the application to iteration of entire functions. Then, in
Section 3, we apply these results to iteration of entire functions. Here, we
use the method of Baker-Domı́nguez described in Subsection 1.1 and further
extend it.
In Section 2, the main results are in Subsections 2.3 and 2.4. In Subsec-
tion 2.3 we prove the following theorem, which is a generalization of Theo-
rems 1.8 and 1.9. For a hyperbolic domain G in the complex sphere Ĉ, λG
always denotes the hyperbolic metric on G.
Theorem 2.24 Let g be an inner function such that λD(gn
(z), gn+1
(z)) →
0 as n → ∞ for some z ∈ D. Then J (g) = ∂D.
Theorem 2.24 will be an easy consequence of two more general theorems
which give necessary and sufficient conditions for an inner function to be
eventually conjugated to a certain Möbius transformation on the Fatou set
and which also classify the possible eventual conjugacies which can arise. See
Subsection 2.1 for the meaning of ‘eventual conjugacy’.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-28-320.jpg)
![ITERATION OF INNER FUNCTIONS 7
of the unit disk or, more generally, of a hyperbolic domain in the complex
sphere. We start with an old theorem of A. Denjoy [15] and J. Wolff [22]; see
also [12, p. 79] or [21, p. 43]. For the notion of the angular limit we refer the
reader to [18, p. 6].
Theorem 2.1 (Denjoy-Wolff). Let h be a non-Möbius holomorphic self-map
of D. Then there is a point p ∈ D such that hn
→ p locally uniformly on D.
Moreover, if p ∈ ∂D, then h has the angular limit p at p.
The point p referred to in this theorem is often called the Denjoy-Wolff
point of h. It may appear that the dynamical behaviour of a holomorphic
self-map depends only on the question whether its Denjoy-Wolff point p is
inside the disk or on its boundary. But, in fact, the case p ∈ ∂D can be
further subdivided. Here we make use of a classification due to C. Cowen
[14] who has shown that a holomorphic self-map of the unit disk is eventually
conjugated to a certain Möbius transformation. Roughly speaking, eventually
conjugated means that the function is semi-conjugated to a Möbius transfor-
mation and, starting at an arbitrary point and iterating, one eventually lands
in a simply connected region where the function is even conjugated to this
Möbius transformation. More precisely, we have the following definition.
Definition 2.2. Let h be a holomorphic self-map of a hyperbolic domain
G ⊂ Ĉ. Let T be a biholomorphic self-map of a simply connected domain
Ω ⊂ C. Then we say that
h ∼ T (h is eventually conjugated to T)
if there exist a holomorphic function Φ : G → Ω and a simply connected
domain V ⊂ G such that the following conditions are satisfied:
(1) Φ ◦ h = T ◦ Φ.
(2) Φ is univalent on V .
(3) V is a fundamental set for h on G, i.e.
h(V ) ⊂ V and ∀ z ∈ G ∃ n ∈ N : hn
(z) ∈ V.
(4) Φ(V ) is a fundamental set for T on Ω.
In this case, (Ω, T, Φ, V ) is called an eventual conjugacy of h on G.
Eventual conjugacies are unique in the following sense; see [14, p. 79-80].
Lemma 2.3 (Cowen). Let h be a holomorphic self-map of a hyperbolic domain
G ⊂ Ĉ. Let (Ω1, T1, Φ1, V1) be an eventual conjugacy of h on G.
(1) Let τ : Ω1 → C be an injective holomorphic function. Then
(τ(Ω1), τ ◦ T1 ◦ τ−1
, τ ◦ Φ1, V1) is an eventual conjugacy of h on G.
(2) Let (Ω2, T2, Φ2, V2) be another eventual conjugacy of h on G. Then
there exists a component W of V1 ∩ V2 such that W is a fundamental
set for h on G and, for each j ∈ {1, 2}, Φj(W) is a fundamental set
for Tj on Ωj. Moreover, there exists a biholomorphic map τ : Ω1 → Ω2
such that Φ2 = τ ◦ Φ1 and T2 = τ ◦ T1 ◦ τ−1
.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-31-320.jpg)
![8 DETLEF BARGMANN
Cowen’s result can be stated as follows. Here, H := {z ∈ C : Im(z) 0}.
Theorem 2.4 (Cowen). Let h be a holomorphic self-map of the unit disk D
without a fixed point. Then exactly one of the following statements holds.
(1) h ∼ idC + 1.
(2) There exists exactly one σ ∈ {−1, 1} such that h ∼ idH + σ.
(3) There exists exactly one λ 1 such that h ∼ λidH.
For the proof of Theorem 2.4, see [14, Theorem 3.2] and the following
remark.
Remark 2.5. By applying Lemma 2.3 it is easy to see that, for each σ ∈
{−1, 1},
h ∼ idH + σ ⇐⇒ h ∼ idσH + 1 ⇐⇒ h ∼ id−iH − iσ
and, for each λ 1,
h ∼ λidH ⇐⇒ h ∼ λid−H ⇐⇒ h ∼ λid−iH.
H. König [17] has given geometrical conditions to determine which Möbius
transformation the function h is eventually conjugated to. Expressing these
conditions in terms of the hyperbolic metric we obtain the following lemma.
Recall that we denote the hyperbolic metric on a hyperbolic domain G ⊂ Ĉ
by λG.
Lemma 2.6. Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ
without a fixed point. For each z ∈ G, let
ρ(z) := inf
n∈N
λG(hn
(z), hn+1
(z)) = lim
n→∞
λG(hn
(z), hn+1
(z)).
Then we have that
(1) h ∼ idC + 1 ⇒ ρ = 0,
(2) h ∼ id±H + 1 ⇒ ρ 0 and inf ρ(G) = 0,
(3) ∃ λ 1 : h ∼ λidH ⇒ inf ρ(G) 0.
Remark 2.7. If additionally G is simply connected, then Cowen’s result im-
plies that all these implications are equivalences. In general, a holomorphic
self-map of a hyperbolic domain need not be semi-conjugated to a Möbius
transformation at all.
To prove Lemma 2.6 we make use of the following lemma.
Lemma 2.8. Let Ω ∈ {C, H, −H} and let T := idΩ + 1. Let W ⊂ Ω be a
fundamental set for T on Ω. Let r 1 and let w ∈ Ω be the center of an
open disk Q with radius r such that Q ⊂ Ω. Then
lim
n→∞
λW (w + n, w + n + 1) ≤
1
2
log
µ
r + 1
r − 1
¶
.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-32-320.jpg)
![10 DETLEF BARGMANN
Lemma 2.9. Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ
such that h is not an automorphism of G and there is a fixed point p of h in
G such that λ := h0
(p) 6= 0. Then |λ| 1 and h ∼ λidC.
Proof. By Montel’s theorem, the family {hn
: n ∈ N} is normal. Since h is
not an automorphism of G we conclude that no subsequence of (hn
)n∈N con-
verges to a non-constant limit function (see for instance [6, Theorem 7.2.4]).
Hence hn
→ p locally uniformly on G, which implies that |λ| 1. Hence
there is an open and connected neighbourhood V of p in G, a number r 0,
and a biholomorphic map φ : V → D(0, r) such that h(V ) ⊂ V , φ(p) = 0,
φ0
(p) = 1 and φ ◦ h|V = λφ. It is easy to see that
Φ(z) :=
1
λn
φ(hn
(z)) if hn
(z) ∈ V
is a well-defined holomorphic function on G such that (C, λidC, Φ, V ) forms
an eventual conjugacy of h on G. 2
Moreover, we make use of the following theorems due to P. Bonfert [11,
Theorem 5.7 and Theorem 6.1].
Theorem 2.10 (Bonfert). Let h be a holomorphic self-map of a hyperbolic
domain G ⊂ C without a fixed point, and without an isolated boundary fixed
point, i.e. there is no isolated boundary point a ∈ ∂G such that h extends
holomorphically to a and fixes a. Suppose that λG(hn
(z), hn+1
(z)) → 0 as
n → ∞ for some z ∈ G. Let z0 ∈ G. Define
φn : G → C, z 7→
hn
(z) − hn
(z0)
hn+1(z0) − hn(z0)
.
Then the sequence (φn)n∈N converges locally uniformly in G to a holomorphic
function φ : G → C such that φ(h(z)) = φ(z) + 1 for all z ∈ G.
Theorem 2.11 (Bonfert). Let T be a Möbius transformation and G ⊂ C be
a hyperbolic domain such that T(G) ⊂ G, T(∞) = ∞, and T has no fixed
point in G. Then
λG(Tn
(z), Tn+1
(z)) → 0 as n → ∞ for (any) z ∈ G
if and only if
[
n∈N
T−n
(G) = C or
[
n∈N
T−n
(G) = C {b},
where b ∈ C G is a fixed point of T.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-34-320.jpg)
![12 DETLEF BARGMANN
(1) g|I is injective.
(2) ηj := limz→ζj
g|I(z) exists, for each j ∈ {1, 2}, and is equal to the
angular limit of g at ζj. Moreover, η1 6= η2, and g(I) = (η1, η2) or
g(I) = (η2, η1).
We make use of the following lemmata to prove Theorem 2.17.
Lemma 2.18. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D be
distinct and let g be holomorphic on I := (ζ, θ). Suppose that
η := lim
z→ζ
g|I(z) and lim
z→θ
g|I(z)
exist and are equal. Then ∂D {η} ⊂ g(I).
Proof. The set Z := g(I) {η} is a non-empty open subset of
∂D {η}. Let w ∈ Z {η}. Then there exists a sequence (xn)n∈N in I
such that limn→∞ g(xn) = w 6= η. We may assume that xn → x ∈ I. Since
limz→ζ g|I(z) = η = limz→θ g|I(z) we conclude that x ∈ I and w ∈ Z. Since
∂D {η} is connected we have that Z = ∂D {η}. 2
Remark 2.19. As a consequence of Lemma 2.18 we obtain that an inner
function g is locally injective at any point ζ ∈ ∂D where it is holomorphic.
This can also be concluded from the Julia-Wolff lemma (see Theorem 2.29).
Lemma 2.20. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D such
that ζ 6= θ and g is holomorphic on I := (ζ, θ). Then either g((ζ, η)) = ∂D,
for each η ∈ I, or limz→ζ g|I(z) exists.
Proof. Suppose that there exists η ∈ I such that g((ζ, η)) 6= ∂D. From
Lemma 2.18 we conclude that g is injective on (ζ, η). Hence g is strictly
orientation preserving or strictly orientation reversing on (ζ, η). In both cases
it is easy to see that limz→ζ g|I(z) exists. 2.
The next result is an easy consequence of the Lehto-Virtanen theorem.
Lemma 2.21. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D such that
ζ 6= θ and g is holomorphic on I := (ζ, θ). Suppose that η := limz→ζ g|I(z)
exists. Then the angular limit of g at ζ exists and equals η.
Proof. Choose a disk D which has its center at a point α ∈ I such that
ζ ∈ ∂D, D∩∂D ⊂ I, and g(D∩∂D) ⊂ D(η; 1). Then D(−η; 1)∩∂D ⊂ Ĉg(D)
which implies that g is a normal function on D. Thus from the Lehto-Virtanen
theorem (see [18], p. 71) we have that there is a path γ : [0, 1) → D ∩ D such
that γ(t) → ζ and γ(t) → η as t → 1. Again we apply the Lehto-Virtanen
theorem and conclude that the angular limit of g at ζ exists and equals η.
2](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-36-320.jpg)
![14 DETLEF BARGMANN
then the conclusion follows from Theorem 1.8. Hence we may assume that
p ∈ ∂D. From Lemma 2.6 we see that g|D ∼ idC + 1.
Assume that p ∈ F(g). Then λ := g0
(p) ∈ (0, 1] and from Lemma 2.9 we
conclude that g|F(g) ∼ λidC. By Theorem 2.22 this implies that g|D ∼ λidH.
This is a contradiction to g|D ∼ idC + 1. Hence p ∈ J (g).
Assume that F(g) ∩ ∂D 6= ∅. From the Schwarz-Pick lemma we conclude
that λF(g)(gn
(z), gn+1
(z)) → 0 as n → ∞. By Theorem 2.23 this implies that
g|D ∼ idH ± 1. This contradicts g|D ∼ idC + 1. Hence J (g) = ∂D. 2
This leads to the following theorem.
Theorem 2.25. Let g be a non-Möbius inner function such that J (g) 6= ∂D.
Suppose that there exists an invariant component of F(g) ∩ ∂D which is not
absorbing. Then there exists λ 1 such that g|D ∼ λidH.
Proof. From Theorem 2.24 we know that g|D 6∼ idC+1. From Theorem 2.23
we conclude that g|D 6∼ idH ± 1. From Theorem 1.8 we see that g does not
have a fixed point in D. Hence the conclusion follows from Theorem 2.4. 2
Remark. An example of a non-Möbius inner function which has an invari-
ant but not absorbing component of F(g) ∩ ∂D is given in Subsection 2.5.
The rest of this subsection is devoted to the proofs of Theorem 2.22
and 2.23. The main part of the proofs is contained in the following lemma.
Lemma 2.26. Let g be a non-Möbius inner function such that J (g) 6= ∂D.
Let p ∈ ∂D be the Denjoy-Wolff point of g. Suppose that there exists an
invariant component I of F(g) ∩ ∂D. Let G := D ∪ I ∪ (Ĉ D). Then exactly
one of the following statements holds.
(1) p ∈ F(g) and g|G ∼ g0
(p)idC and g|D ∼ g0
(p)idH.
(2) p ∈ J (g) and g|G ∼ idC + 1 and g|D ∼ idH ± 1.
(3) p ∈ J (g) and there exists λ 1 such that g|G ∼ λidH and g|D ∼ λ2
idH.
Proof. By conjugating with an appropriate transformation we may assume
that g is an inner function of the upper half plane H with Denjoy-Wolff point
p = ∞ and that I is an invariant component of F(g)∩(R∪{∞}). Since J (g)
is a perfect set the simply connected domain G := H ∪ I ∪ −H is hyperbolic.
By Theorem 2.4 and Lemma 2.9 there exist
(Ω, T) ∈ {(C, idC + 1)}
∪ {(−iH, id−iH + iσ) : σ ∈ {−1, 1}}
∪ {(−iH, λid−iH) : λ 1}
∪ {(C, g0
(p)idC},
a simply connected domain V ⊂ G, and a holomorphic function Φ : G → Ω
such that (Ω, T, Φ, V ) forms an eventual conjugacy of g on G. Here (Ω, T) =
(C, g0
(p)idC) if and only if ∞ = p ∈ I. By Lemma 2.13 this is the case if and
only if p ∈ F(g).](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-38-320.jpg)
![ITERATION OF INNER FUNCTIONS 17
The next lemma is proved in [1, p. 116, Lemma 2].
Lemma 2.28. Let γ be a closed rectifiable curve in C such that γ−1
(H) 6=
∅ 6= γ−1
(−H) and γ−1
(R) consists of exactly two points a and b. Then the
open interval with endpoints γ(a) and γ(b) is contained in int(γ).
Now, we are able to prove Theorem 2.22 and Theorem 2.23.
Proof of Theorem 2.22.
(2) ⇒ (1) and ((4) or (5) or (6)). Suppose that there exists an absorb-
ing component I of F(g) ∩ ∂D. By Lemma 2.14, the set I is invariant under
g. Let G := D ∪ I ∪ (Ĉ D). By Lemma 2.26, exactly one of the following
statements holds:
(40
) p ∈ F(g) and g|G ∼ g0
(p)idC and g|D ∼ g0
(p)idH,
(50
) p ∈ J (g) and g|G ∼ idC + 1 and g|D ∼ idH ± 1,
(60
) p ∈ J (g) and there exists λ 1 such that g|G ∼ λidH and g|D ∼ λ2
idH.
Since I is absorbing we conclude from Lemma 2.27 that (40
) implies (4), (50
)
implies (5), and (60
) implies (6).
(3) ⇒ (2). Suppose that there are no wandering components of F(g)∩∂D.
Since F(g) ∩ ∂D 6= ∅ we conclude from Lemma 2.14 that there exists an
invariant component I of F(g)∩∂D. Let J be a component of F(g)∩∂D. By
Lemma 2.14 there exists n ∈ N such that gn
(J) is contained in an invariant
component of F(g) ∩ ∂D. By Lemma 2.13 there is at most one invariant
component which implies that gn
(J) ⊂ I. Hence I is an absorbing component
of F(g) ∩ ∂D.
(1) ⇒ (3). Suppose that there exists an eventual conjugacy (Ω, T, Φ, V ) of
g on F(g). We may assume that p ∈ J (g) because otherwise the component
I of p in F(g) ∩ ∂D is absorbing and the conclusion follows. By conjugating
with an appropriate transformation we may further assume that g is an inner
function of H and p = ∞. Let I be a component of F(g) ∩ R. Let x ∈ I and
let γ be the positively oriented circle centered at (x + g(x))/2 which contains
x and g(x). Then γ is a compact subset of F(g) and there exists n ∈ N such
that gn
(γ) ⊂ V . Let J be the component of F(g) ∩ R which contains gn
(x).
Since V is simply connected and gn
◦ γ is a closed analytic curve in V we
conclude that int(gn
◦γ) ⊂ V . From Lemma 2.28 we see that the open interval
K with endpoints gn
(x) and gn+1
(x) is contained in int(gn
◦ γ) ⊂ V ⊂ F(g).
Hence K ⊂ J and {gn
(x), gn+1
(x)} ⊂ J. This implies that I is not wandering.
2
Proof of Theorem 2.23.
(1) ⇒ (2). Suppose that g|F(g) ∼ idC + 1. Then g does not have a fixed
point in F(g) which implies that p ∈ J (g). From Lemma 2.6 we know that
λF(g)(gn
(z), gn+1
(z)) → 0 as n → ∞ for every z ∈ F(g).
(2) ⇒ (3). Suppose that p ∈ J (g) and λF(g)(gn
(z), gn+1
(z)) → 0 as
n → ∞ for some z ∈ F(g). By conjugating appropriately we may assume](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-41-320.jpg)
![18 DETLEF BARGMANN
that g is an inner function of H and p = ∞. Since J (g) is perfect, F(g) is a
hyperbolic domain in C such that F(g) has no isolated boundary points. Let
x ∈ F(g) ∩ R and define
φn : F(g) → C, z 7→
gn
(z) − gn
(x)
gn+1(x) − gn(x)
= Tn(gn
(z)),
where Tn(w) := (w − gn
(x))/(gn+1
(x) − gn
(x)). From Theorem 2.10 we know
that the sequence (φn)n∈N converges locally uniformly on F(g) to a holomor-
phic function φ : F(g) → C such that φ(g(z)) = φ(z) + 1 for all z ∈ F(g).
Let G := φ(F(g)). We first prove that
(∗)
[
n∈N
(G − n) = C.
Proof of (∗). We may assume that G is hyperbolic because otherwise CG
contains at most one point and (∗) follows immediately. Since φ◦g|F(g) = φ+1
we have that (idC + 1)(G) ⊂ G. Moreover, from the Schwarz-Pick lemma we
conclude that
λG(φ(z) + n, φ(z) + n + 1) = λG(φ(gn
(z), φ(gn+1
(z)))
≤ λF(g)(gn
(z), gn+1
(z))
→ 0
as n → ∞ for some z ∈ F(g). Thus we can apply Theorem 2.11 and obtain
that (∗) holds.
Since G+1 ⊂ G, the sequence (G−n)n∈N is increasing, which implies that
there exists k ∈ N such that [0, 1] ⊂ G−k, and hence [φ(gk
(x)), φ(gk+1
(x))] =
[k, k + 1] ⊂ G ∩ R = φ(F(g)) ∩ R. Since φn → φ locally uniformly in F(g)
we find n0 ∈ N such that, for each n ≥ n0,
[φn(gk
(x)), φn(gk+1
(x))] ⊂ φn(F(g)) ∩ R = φn(F(g) ∩ R).
Hence, for each n ≥ n0,
[Tn(gn+k
(x), Tn
(gn+k+1
(x))] ⊂ Tn(F(g) ∩ R),
which implies that the closed interval with endpoints gk+n
(x) and gk+n+1
(x) is
contained in F(g) ∩ R for every n ≥ n0. Let I be the component of F(g) ∩ R
which contains u := gk+n0
(x). Then g(u) ∈ I and hence g(I) ⊂ I which
implies that I is unbounded because (g|F(g))n
→ ∞ as n → ∞. Moreover,
g|I does not have any fixed point which implies that either g(z) z for each
z ∈ I and sup I = ∞ or g(z) z for each z ∈ I and inf I = −∞.
We show that I is an absorbing component of F(g) ∩ R. To this end let
y ∈ F(g)∩R. Then φn(y) ∈ R, for each n ∈ N, and hence φ(y) ∈ R. Thus we
find m ∈ N such that φ(gm
(y)) = φ(y)+m 1. Hence there exists n ≥ n0 +k
such that
gn
(gm
(y)) − gn
(x)
gn+1(x) − gn(x)
= φn(gm
(y)) ≥ 1.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-42-320.jpg)
![ITERATION OF INNER FUNCTIONS 19
Let l := n − n0 − k. If now g(z) z, for each z ∈ I, then
gn+1
(x) = gl+1
(u) gl
(u) = gn
(x),
which implies that gn
(gm
(y)) gn
(x) = gl
(u) and hence gn+m
(y) ∈ I because
sup I = ∞. We can argue analogously for the case when g(z) z for each
z ∈ I. Thus gn+m
(y) ∈ I and I is absorbing.
It remains to show that gH ∼ id±H + 1. To this end, first note that
φn(H) ⊂ H or φn(H) ⊂ −H for each n ∈ N. Since φ is not constant we
conclude that there exists σ ∈ {−1, 1} such that φ(H) ⊂ σH. According
to Theorem 2.4 there exists a fundamental set V ⊂ H for g on H. Since
g(V ) ⊂ V and g|V is injective we have that φn|V is injective, for each n ∈ N,
and hence φ is injective on V . Since
φ(g(z)) = φ(z) + 1 for every z ∈ H
it remains to show that W := φ(V ) is a fundamental set for idσH + 1. Let
w ∈ σH. According to (∗) there exists q ∈ N such that w+q ∈ G∩σH = φ(H).
Choose z ∈ H such that w + q = φ(z). Since V is a fundamental set for g on
H there exists r ∈ N such that gr
(z) ∈ V and hence
w + q + r = φ(z) + r = φ(gr
(z)) ∈ φ(V ) = W.
Thus W is a fundamental set for idσH + 1 and g|H ∼ idσH + 1. This proves
(3).
The implication (3) ⇒ (4) follows immediately from Lemma 2.14. To
prove (4) ⇒ (1) suppose that g|D ∼ idH ± 1 and there exists an invariant
component I of F(g) ∩ ∂D. Let G := D ∪ I ∪ (Ĉ D). From Lemma 2.26
we conclude that p ∈ J (g) and g|G ∼ idC + 1. By Lemma 2.6 we have that
λG(gn
(z), gn+1
(z)) → 0 as n → ∞, for each z ∈ G, which by the Schwarz-
Pick lemma implies that λF(g)(gn
(z), gn+1
(z)) → 0 as n → ∞ for every z ∈ G.
Thus (2) holds and we conclude from the implication (2) ⇒ (3) that there
exists an absorbing component of F(g)∩∂D. Since each absorbing component
is invariant and there is at most one invariant component of F(g) ∩ ∂D we
have that I is absorbing. Thus we can apply Lemma 2.27 and obtain that
g|F(g) ∼ idC + 1. 2
2.4. Periodic points of an inner function. It is well-known that the Julia
set of a non-injective rational, entire, or meromorphic function f is equal to
the closure of the set of repelling periodic points of f. In this subsection we
see that the same is true for inner functions if the definition of a repelling
periodic point is relaxed in a certain sense.
We make use of the following theorem which is known as the Julia-Wolff
lemma [18, Proposition 4.13]. For the definition and the main properties of
the angular derivative we refer the reader to [18].](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-43-320.jpg)
![20 DETLEF BARGMANN
Theorem 2.29 (Julia-Wolff). Let h be a holomorphic self-map of the unit
disk. Let ζ, η ∈ ∂D such that h has the angular limit η at ζ. Then the angular
derivative h0
(ζ) exists in (0, ∞] and satisfies
h0
(ζ) =
η
ζ
sup
z∈D
1 − |z|2
|ζ − z|2
|η − h(z)|2
1 − |h(z)|2
.
Definition 2.30. Let h be a holomorphic self-map of the unit disk. Let
ζ ∈ ∂D.
(1) We call ζ a fixed point of h if h has the angular limit ζ at ζ. In this
case the fixed point ζ of h is called repelling if the angular derivative
h0
(ζ) is contained in (1, ∞].
(2) We call ζ a (repelling) periodic point of h if there exists n ∈ N such
that ζ is a (repelling) fixed point of hn
.
Theorem 2.1 represents only one part of the theorem of Denjoy and Wolff.
The second part of the Denjoy-Wolff theorem (see [15] and [22]) is a statement
about the angular derivatives at the fixed points of a holomorphic self-map
of D:
Theorem 2.31 (Denjoy-Wolff). Let h be a non-Möbius holomorphic self-map
of the unit disk. Let p ∈ D be the Denjoy-Wolff point of h. Then p is the
unique fixed point of h which is not repelling.
Hence there is at most one periodic point of a non-Möbius inner function
g which is not repelling. Since J (g) is perfect it remains to show that the
periodic points of g are dense in the Julia set. This is a consequence of the
following theorem.
Theorem 2.32. Let g be an inner function. Suppose that ζ ∈ ∂D is a
singularity of g. Then ζ is an accumulation point of fixed points of g.
We use the following lemma to prove Theorem 2.32.
Lemma 2.33. Suppose that g is a non-Möbius inner function of the upper
half plane H. Then the following statements are equivalent.
(1) ∞ is the Denjoy-Wolff point of g.
(2) Im(g(z)) ≥ Im(z), for each z ∈ H.
(3) G(z) := g(z) − z is an inner function of the upper half plane.
(4) Im(g(z)) Im(z), for each z ∈ H.
Proof. Suppose that (1) holds. Let T(z) := i(1 + z)/(1 − z). Then
h := T−1
◦ g ◦ T is an inner function of the unit disk with Denjoy-Wolff point
1. An easy calculation shows that
Im(T(z)) =
(1 − |z|)2
|1 − z|2](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-44-320.jpg)
![ITERATION OF INNER FUNCTIONS 21
for each z ∈ D. Since the angular derivative h0
(1) is contained in (0, 1] we
conclude from Theorem 2.29 that Im(g(z)) ≥ Im(z), for each z ∈ H.
Suppose that (2) holds. Let G(z) := g(z) − z. Then Im(G(z)) ≥ 0, for
each z ∈ H, which implies that G is an inner function of H or G is constant.
Since g is non-Möbius we conclude that G is non-constant and hence an inner
function. The implication (3) ⇒ (4) is trivial. Suppose that (4) holds. It is
easy to see that, for each p ∈ H ∪ R, there exists z ∈ H such that (gn
(z))n∈N
does not converge to p. Hence we conclude from the Denjoy-Wolff theorem
that ∞ is the Denjoy-Wolff point of g. 2
Proof of Theorem 2.32. Let p ∈ D be the Denjoy-Wolff point of g.
Case 1. Suppose that p ∈ D. By conjugating with a Möbius transformation
if necessary we may assume that p = 0. It is easy to see that
h(z) :=
g(z)
z
is an inner function with a singularity at ζ. Let U be a neighbourhood of ζ.
By Lemma 1.5 there exists θ ∈ (U ∩ ∂D) {ζ} and a path γ : [0, 1) → D such
that limr→1 γ(r) = θ and h(γ(r)) → 1 as r → 1 in a Stolz angle. Hence we
have that g(γ(r)) → θ as r → 1. By the Lehto-Virtanen theorem [18, p. 71],
we conclude that g has angular limit θ at θ.
Case 2. Suppose that p ∈ ∂D. We may assume that p = 1. Let T(z) :=
i(1 + z)/(1 − z). Then G := T ◦ g ◦ T−1
is an inner function of the upper half
plane with Denjoy-Wolff point ∞ and a singularity at T(ζ). From Lemma 2.33
we conclude that H(z) := G(z) − z is an inner function of the upper half
plane which has a singularity at T(ζ). Let U be a neighbourhood of T(ζ).
By Lemma 1.5 there exists θ ∈ (U ∩ R) {ζ} and a path γ : [0, 1) → H such
that limr→1 γ(r) = θ and H(γ(r)) → 0 as r → 1 in a Stolz angle. Hence we
have that G(γ(r)) → θ as r → 1. By the Lehto-Virtanen theorem again, we
conclude that G has angular limit θ at θ. 2
Theorem 2.34. Let g be a non-Möbius inner function. Then J (g) is the
closure of the set of the repelling periodic points of g.
Proof. For the case when g is a finite Blaschke product this is an old
theorem of G. Julia and P. Fatou; see for instance [6, Theorem 6.9.1] or [20], [5]
and [10] for simpler proofs.
For the case when g has a singularity on ∂D it follows from Theorem 1.4
and Theorem 2.32 that J (g) is contained in the closure of the set of the
periodic points of g. Since there is at most one periodic point of g which is
not repelling and J (g) is perfect we conclude that even the repelling periodic
points of g are dense in J (g).
On the other hand, the only periodic point of g which might be in F(g) is
the Denjoy-Wolff point. Hence all repelling periodic points are contained in
J (g). 2.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-45-320.jpg)
![24 DETLEF BARGMANN
Example 2.38. The function
g(z) := 2z −
X
n∈N
2−n
z − 2n
is an inner function of the upper half plane with the following property: g has
a wandering and an invariant component of F(g)∩(R∪{∞}). In particular,
the invariant component is not absorbing.
Proof. Since Q := {2n
: n ∈ N} is bounded from the left there exists
c ∈ (0, ∞) such that Re(g(z)) −c whenever Re(z) −c. This implies that
there exists an invariant component I of F(g) ∩ (R ∪ {∞}). Since ∞ is a
accumulation point of Q we conclude that ∞ ∈ J (g).
Choose 1 a b 2 such that
x(x − 1) ≥
1
2
and x(2 − x) ≥
1
2
,
for each x ∈ [a, b]. We prove:
(∗) ∀ a ≤ α β ≤ b, ∀ m ∈ N, ∀ x ∈ [2m
α, 2m
β] :
2m+1
α(1 − 2−m
) ≤ g(x) ≤ 2m+1
β(1 + 2−m
).
Proof of (∗). Let a ≤ α β ≤ b, m ∈ N, and x ∈ [2m
α, 2m
β]. Then
−
X
n∈N
2−n
x − 2n
≤
X
n≥m+1
2−n
2n − x
≤
X
n≥m+1
2−n
2n − 2mβ
= 4−m
X
n∈N
2−n
2n − β
≤ 4−m
(2 − β)−1
.
This implies that
g(x) ≤ 2x + 4−m
(2 − β)−1
≤ 2m+1
β + β4−m
(β(2 − β))−1
≤ 2m+1
β(1 + 2−m
).
On the other hand,
−
X
n∈N
2−n
x − 2n
≥ −
X
n≤m
2−n
x − 2n
≥ −
X
n≤m
2−n
2mα − 2n
≥ −
X
n≤m
2−n
2mα − 2m
≥ −2−m
(α − 1)−1
,
which implies that
g(x) ≥ 2x − 2−m
(α − 1)−1
≥ 2m+1
α − α2−m
(α(α − 1))−1
≥ 2m+1
α(1 − 2−m
).
Thus we have proved (∗). Since
Q
m∈N(1 − 2−m
) and
Q
m∈N(1 + 2−m
) are
convergent, we can find a α0 β0 b and m0 ∈ N such that
α0
Y
m≥m0
(1 − 2−m
) ≥ a and β0
Y
m≥m0
(1 + 2−m
) ≤ b.](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-48-320.jpg)
![This ebook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg License included with this
ebook or online at www.gutenberg.org. If you are not located in the
United States, you will have to check the laws of the country where
you are located before using this eBook.
Title: Scrambled World
Author: Basil Wells
Release date: December 29, 2020 [eBook #64172]
Most recently updated: October 18, 2024
Language: English
Credits: Greg Weeks, Mary Meehan and the Online Distributed
Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK SCRAMBLED
WORLD ***](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-54-320.jpg)
![By BASIL WELLS
[Transcriber's Note: This etext was produced from
Planet Stories Spring 1947.
Extensive research did not uncover any evidence that
the U.S. copyright on this publication was renewed.]](https://image.slidesharecdn.com/22161-250511131833-c8a5fe9a/85/Transcendental-Dynamics-and-Complex-Analysis-1st-Edition-Philip-J-Rippon-57-320.jpg)
Transcendental Dynamics and Complex Analysis 1st Edition Philip J. Rippon
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- 6. 255 Symmetries and integrability of difference equations, P.A. CLARKSON & F.W. NIJHOFF (eds) 256 Aspects of Galois theory, H. VÖLKLEIN et al 257 An introduction to noncommutative differential geometry and its physical applications 2ed, J. MADORE 258 Sets and proofs, S.B. COOPER & J. TRUSS (eds) 259 Models and computability, S.B. COOPER & J. TRUSS (eds) 260 Groups St Andrews 1997 in Bath, I, C.M. CAMPBELL et al 261 Groups St Andrews 1997 in Bath, II, C.M. CAMPBELL et al 262 Analysis and logic, C.W. HENSON, J. IOVINO, A.S. KECHRIS & E. ODELL 263 Singularity theory, B. BRUCE & D. MOND (eds) 264 New trends in algebraic geometry, K. HULEK, F. CATANESE, C. PETERS & M. REID (eds) 265 Elliptic curves in cryptography, I. BLAKE, G. SEROUSSI & N. SMART 267 Surveys in combinatorics, 1999, J.D. LAMB & D.A. PREECE (eds) 268 Spectral asymptotics in the semi-classical limit, M. DIMASSI & J. SJÖSTRAND 269 Ergodic theory and topological dynamics, M.B. BEKKA & M. 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OGDEN (eds) 284 Foundations of computational mathematics, R. DEVORE, A. ISERLES & E. SÜLI (eds) 285 Rational points on curves over finite fields, H. NIEDERREITER & C. XING 286 Clifford algebras and spinors 2ed, P. LOUNESTO 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE et al 288 Surveys in combinatorics, 2001, J. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE 290 Quantum groups and Lie theory, A. PRESSLEY (ed) 291 Tits buildings and the model theory of groups, K. TENT (ed) 292 A quantum groups primer, S. MAJID 293 Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK 294 Introduction to the theory of operator spaces, G. PISIER 295 Geometry and integrability, L. MASON & Y. NUTKU (eds) 296 Lectures on invariant theory, I. DOLGACHEV 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 298 Higher operads, higher categories, T. LEINSTER 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Discrete and continuous nonlinear Schrödinger systems, M.J. ABLORWITZ, B. PRINARI & A.D. TRUBATCH 303 Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford Vol. 1, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford Vol. 2, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 306 Peyresq lectures on geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed) 308 Topology, geometry and quantum field theory, U.L. TILLMANN (ed) 309 Corings and comdules, T. BRZEZINSKI & R. WISBAUER 310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) 311 Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al (eds) 313 Transcendantal aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD et al (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI, N. SMART 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 328 Fundamentals of hyperbolic manifolds, R.D. CANARY, A. MARDEN, & D.B.A. EPSTEIN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L. PARDO, A. PINKUS, E. SULI & M. TODD (eds) 332 Handbooks of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry 2ed, A. KOCK 334 The Navier-Stokes equations, P.G. DRAZIN & N. RILEY 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 Vol. 1, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 Vol. 2, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives Vol. 1, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives Vol. 2, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics, 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds)
- 7. London Mathematical Society Lecture Note Series: 348 Transcendental Dynamics and Complex Analysis Edited by PHILIP J. RIPPON The Open University GWYNETH M. STALLARD The Open University A Tribute to Noel Baker
- 8. cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9780521683722 C Cambridge University Press 2008 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2008 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Transcendental dynamics and complex analysis / edited by Philip J. Rippon, Gwyneth M. Stallard. p. cm. – (London Mathematical Society lecture note series; 348) Includes bibliographical references and index. ISBN 978-0-521-68372-2 (pbk.) 1. Functions of complex variables. 2. Differentiable dynamical systems. 3. Mathematical analysis. I. Rippon, P. J. II. Stallard, Gwyneth M. QA331.7.T73 2008 515.9–dc22 2007050517 ISBN 978-0-521-68372-2 (paperback) Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
- 9. Contents Preface vii Introduction ix 1 Iteration of inner functions and boundaries of components of the Fatou set 1 D. Bargmann 2 Conformal automorphisms of finitely connected regions 37 A. F. Beardon and D. Minda 3 Meromorphic functions with two completely invariant domains 74 W. Bergweiler and A. Eremenko 4 A family of matings between transcendental entire functions and a Fuchsian group 90 S. Bullett and M. Freiberger 5 Singular perturbations of zn 111 R. L. Devaney, M. Holzer, D. M. Look, M. Moreno Rocha and D. Uminsky 6 Residual Julia sets of rational and transcendental functions 138 P. Domı́nguez and N. Fagella 7 Bank-Laine functions via quasiconformal surgery 165 D. Drasin and J. K. Langley 8 Generalisations of uniformly normal families 179 W. K. Hayman and A. Hinkkanen 9 Entire functions with bounded Fatou components 187 A. Hinkkanen 10 On multiply connected wandering domains of entire functions 217 M. Kisaka and M. Shishikura 11 Fractal measures and ergodic theory of transcendental meromorphic functions 251 J. Kotus and M. Urbański 12 Combinatorics of bifurcations in exponential parameter space 317 L. Rempe and D. Schleicher 13 Baker domains 371 P. J. Rippon 14 Escaping points of the cosine family 396 G. Rottenfusser and D. Schleicher 15 Dimensions of Julia sets of transcendental meromorphic functions 425 G. M. Stallard 16 Abel’s functional equation and its role in the problem of croissance régulière 447 G. Szekeres v
- 10. Professor Noel Baker (1932–2001) vi
- 11. PREFACE This book was written in honour of Noel Baker following his sudden death in 2001. It comprises a collection of articles written by friends, colleagues and former students of Noel. In particular, we are delighted that Noel’s MSc supervisor and long-time friend, George Szekeres, was able to contribute a paper to this volume — he sadly died before the book was published. All of these articles deal with topics that interested Noel and, in most cases, they are in areas where Noel’s own work has been very influential. Several of the papers are survey articles that we hope will be a valuable addition to the literature. There are also new results that Noel would have been delighted to have seen. Most of the papers deal with the iteration of transcendental meromorphic functions — the field in which Noel was pre-eminent and in which he carried out much of the pioneering work — and there are also some papers in closely related topics that he would have enjoyed. As this volume shows, much of the recent work in complex dynamics (as the subject of iteration theory is now called) builds on ideas and techniques that Noel introduced and that will continue to be used by all those who work in this field. We hope that this book will be a fitting memorial to a man who inspired so many of us. Phil Rippon and Gwyneth Stallard Department of Mathematics and Statistics The Open University Milton Keynes MK7 6AA vii
- 13. INTRODUCTION In this introduction, we summarise the mathematical career of Noel Baker and indicate how the papers in this volume relate to his work. Much of the material is taken from the obituary of Noel Baker that appeared in the Bulletin of the London Mathematical Society [17]. Noel Baker was born on 10 August 1932 and died, of a heart attack, on 20 May 2001. He grew up in Australia and was first introduced to the theory of iteration by his MSc supervisor, George Szekeres, who suggested that he work on the functional equation f(f(z)) = F(z), where f and F are analytic functions. In his first mathematical paper (1), Noel used the theory of iteration of analytic functions, which had been devel- oped principally by Fatou and Julia and which was not well known at that time. He used this theory to show, amongst other things, that if F belongs to a certain class of entire functions, which includes the exponential function, then the above equation has no entire solution. This first paper also contains examples that were constructed using Wiman-Valiron theory. Throughout his career Noel was to find ever more techniques from classical complex analysis that can usefully be applied to iteration theory. In 1955 Noel won a German government scholarship to the University of Tübingen, where he worked under Hellmuth Kneser. Noel’s doctoral thesis, published in (2), continued his study of functional equations. From 1957 to 1959, Noel taught mathematics at the University of Alberta in Edmonton, Canada. In 1959 he moved to Imperial College London, where he remained until retirement in 1997. In his research, Noel worked on many problems in complex analysis and had a wide range of collaborators, but iteration theory, his great love, was for many years a lone interest. However, when the subject was reborn around 1980, partly as a result of the advent of accessible computer graphics, it became clear to the new adherents that Noel had for many years been quietly and carefully completing the foundations begun earlier in the century by the French mathematicians Pierre Fatou and Gaston Julia. He had also pointed the way towards many future developments, both by proving new results and by posing challenging problems. In the explosion of research on iteration theory that took place in the subsequent years, many of the papers published on iteration made reference to Noel’s work and he received many invitations to speak at international conferences on iteration. At these he would often appear reserved, much preferring to let others speak about the latest work, ix
- 14. x INTRODUCTION even though he was the acknowledged authority on many matters, and the person whose judgement about the validity of a new proof was always sought. Noel continued his research after his retirement and one of his last papers was dedicated to George Szekeres on the occasion of the latter’s 90th birthday. Noel’s early work on functional equations led him to consider problems about periodic points, which play a very important role in complex dynamics. It was already known that for an entire function there must be infinitely many periodic points of period p, for all p ≥ 2, but Noel considered the unsolved problem of the existence of periodic points of a given exact period. He showed in (6) that for all non-linear entire functions there exist periodic points of exact period p, for all p with at most one exception; for example, f(z) = z+ez has no fixed points. In a later paper (13) Noel showed that for a polynomial the only possible exceptional value in this result is p = 2, the corresponding exceptional functions being f(z) = z2 − z and other quadratics ‘similar’ to this one. He also conjectured that for a transcendental entire function the only possible exceptional value is p = 1, and this was proved by Bergweiler [2]. We now describe the origins of complex dynamics. Let f be a rational function of degree at least 2 or a transcendental entire function. The set of points near which the sequence of iterates fn forms a normal family is called the Fatou set F(f) and its complement is called the Julia set J(f). Roughly speaking, the dynamics are stable on the Fatou set and chaotic on the Julia set. Also, the Julia set often exhibits great topological complexity as well as ‘self-similarity’; for example, the paper in this volume by Devaney et al discusses a family of rational functions whose Julia sets in some cases contain Cantor sets of curves and in other cases contain Sierpinski curves. The fundamental properties of the sets F(f) and J(f) were first established for rational functions in [13] and [8], and for transcendental entire functions in [9]. In the last paper, Fatou studied the iteration of transcendental entire functions in some detail, giving examples that pointed to significant differ- ences to the theory that had been developed for rational functions. He asked the following fundamental questions about a transcendental entire function f: 1. Are the repelling periodic points of f dense in J(f)? 2. Are there examples where J(f) = C? In particular, is this true for f(z) = ez ? 3. Can J(f) be totally disconnected? 4. Must J(f) contain infinitely many unbounded analytic curves, at each point of which fn → ∞? Question 1 is of great theoretical importance, and it had been answered ‘yes’ for rational functions by both Fatou and Julia. Fatou had also given an example of a rational function f for which J(f) is totally disconnected, and Lattès [14] an example for which J(f) = C. Most of Fatou’s questions were solved by Noel during the decade 1965–1975, as we now indicate.
- 15. INTRODUCTION xi The first question was answered in the affirmative in the paper (22), which is of fundamental importance in complex dynamics and appropriately dedicated to Hellmuth Kneser. Here, Noel called on a deep covering theorem due to Ahlfors (see [11, page 148]) to show that arbitrarily close to each point of J(f) there is a repelling periodic point of f. From this, he deduced the general result that if f is any non-linear entire function, then the set of entire functions that commute with f is countable. Many authors have tried to simplify the proof in (22) that the repelling periodic points are dense in J(f), in order to avoid the deep theorem of Ahlfors. Eventually, more elementary proofs based on a renormalisation technique were given by Schwick [18], Bargmann [1], and Berteloot and Duval [4]. Two years later, in (25), Noel answered the first part of Fatou’s second ques- tion by showing that there is a function of the form f(z) = kzez , where k 0, such that J(f) = C. A proof that if f(z) = ez , then we have J(f) = C was given ten years later by Misiurewicz [15]. Noel answered Fatou’s third question in the negative in (32). If J(f) is totally disconnected, then F(f) must have a single unbounded multiply connected component. Noel had already constructed in (9) an example of a transcen- dental entire function for which F(f) has at least one multiply connected component. This function was of the form f(z) = Cz2 ∞ Y n=1 1 + z rn , in which the positive constants r1 r2 . . . have the property that f(An) ⊂ An+1, where An = {z : r2 n |z| r 1/2 n+1}. However, Noel did not determine in (9) whether F(f) has a single unbounded multiply connected component or a sequence of bounded multiply connected components. In (33) he used Schottky’s theorem [11, page 169], yet another result from classical complex analysis, to show that the latter must be the case. This solved another important problem in complex dynamics, open since the work of Fatou and Julia, by showing that the above function has a sequence of wandering domains, that is, distinct components Un of F(f) such that f(Un) ⊂ Un+1, for n = 1, 2, . . . . In contrast, Sullivan [16] showed that rational functions do not have wandering domains. The paper (32), written later than but published earlier than (33), used Schottky’s theorem once again to show that a transcendental entire function cannot have an unbounded multiply connected component of F(f), thus proving that J(f) can never be totally disconnected. The results in (32) and (33) led to much further work. In (53), Noel showed that wandering domains for transcendental entire functions may be infinitely connected. For many years it was not known whether such wandering domains
- 16. xii INTRODUCTION could be finitely connected. In this volume, Kisaka and Shishikura show that they can in fact have any given finite connectivity. The result in (32) shows that if f is a transcendental entire function, then J(f) must contain a continuum, so its Hausdorff dimension dimHJ(f) is at least 1. It remains an open question whether dimHJ(f) = 1 is possible. In this volume, there is a survey article on dimensions of Julia sets by Stallard, complemented by a survey article on fractal measures and ergodic theory by Kotus and Urbański. Noel’s wandering domains example mentioned earlier shows that the answer to Fatou’s fourth question (as stated here) is ‘no’. However, the structure of the ‘escaping set’, where fn → ∞, continues to stimulate much work, includ- ing the paper by Rottenfusser and Schleicher that appears in this volume. Sullivan’s remarkable result [16] that rational functions do not have wan- dering domains was proved using new techniques based on quasiconformal conjugacy. Noel quickly saw that these new techniques would also apply to various families of transcendental entire functions, and a proof that exponen- tial functions have no wandering domains appeared in (49). This was one of a number of papers at that time that established many of the basic dynamical properties of the exponential family and began the description of the corre- sponding parameter space, the ‘exponential Mandelbrot set’, which has since been the subject of much study — see, for example, the paper by Rempe and Schleicher in this volume. In (41), Noel initiated another major development by showing that if a tran- scendental entire function f has order of growth at most 1/2, minimal type, then F(f) has no unbounded invariant components, and he also gave a more restrictive condition on the maximum modulus of f that forces every compo- nent of F(f) to be bounded. The question of whether the latter conclusion follows from order at most 1/2, minimal type, remains open, though many authors have obtained partial results in this direction; this volume contains a survey article on this problem by Hinkkanen. A key step in Noel’s proof in (41) is to exclude unbounded invariant com- ponents of F(f) in which fn → ∞. He did this by establishing estimates for the growth of iterates in such components, which he later refined in (57). In recognition of his work on Fatou components of this type, Eremenko and Lyubich introduced the name Baker domain for such components in [7]. In this fundamental paper, Eremenko and Lyubich showed that if the set S(f) of inverse function singularities of a transcendental entire function f is bounded, then f has no Baker domains and if S(f) is finite, then f has no wandering domains; see also [10]. A survey article on Baker domains by Rippon appears in this volume. Yet another fundamental contribution to the iteration of transcendental entire functions came in the papers (65), (73) and (74). Once again an unbounded
- 17. INTRODUCTION xiii invariant component U of F(f) was considered, but now the aim was to describe the nature of the boundary of U. Some special cases had been investigated by other authors, following the appearance of computer pictures of Julia sets, but Noel and his students Weinreich and Domı́nguez attacked the general case. In (65), it was shown that • if U is not a Baker domain (that is, U is an attracting basin, a para- bolic basin, or a Siegel disc), then ∂U is sufficiently complicated that ∞ belongs to the impression of every prime end of U; • if ∂U is a Jordan curve in the extended complex plane Ĉ (and such U do exist), then not only must U be a Baker domain, but f must be univalent in U. The key tool introduced in this work arises from the fact that if Ψ is a conformal map from the unit disc D onto U, then Ψ−1 ◦ f ◦ Ψ is an inner function, that is, an analytic self-map of D whose angular limits have modulus 1 almost everywhere on ∂D. The paper (65) initiated a version of Fatou-Julia theory for inner functions, a topic now of interest in its own right, and this theory was taken further in (73). Further results on this theory are given in the paper by Bargmann in this volume. Many of Noel’s final papers are joint papers with his last student, Domı́nguez, and concern the connectedness properties of the Julia set. Many of these results are described and extended in the paper by Domı́nguez and Fagella in this volume. Fatou-Julia theory of the iteration of general transcendental meromorphic functions was established in the fundamental papers (62), (63), (64) and (66) by Baker, Kotus and Lü. The Fatou set F(f) is here taken to be the set of points near which the iterates fn are defined and form a normal family, and then J(f) = Ĉ F(f). Many of the basic results turn out to be similar to those for rational and entire functions, but there are some striking differ- ences. For example, in (62) the authors showed that J(f) is once again the closure of the repelling periodic points of f, and this fact is used to give a complete classification of those transcendental meromorphic functions, such as f(z) = tan z, for which J(f) is a subset of the real line; there are no tran- scendental entire functions for which the Julia set is contained in the real line. Then, in (63), they used techniques from approximation theory, pioneered by Eremenko and Lyubich [6], to construct transcendental meromorphic func- tions with wandering domains of all possible connectivities. The question of periodic components was taken up in (64), where the authors showed that precisely five possible types can arise for a transcendental mero- morphic function, namely, attracting basins, parabolic basins, Siegel discs, Herman rings and Baker domains. Moreover, any invariant components of F(f) must be simply connected, doubly connected, or infinitely connected.
- 18. xiv INTRODUCTION But perhaps the most striking result here was the construction of a tran- scendental meromorphic function f with a preperiodic component of F(f) of any given finite connectivity. This construction used the powerful tech- nique of quasiconformal surgery, introduced by Shishikura [19], which also appears in many of the papers in this volume — namely, those by Drasin and Langley, Domı́nguez and Fagella, and Kisaka and Shishikura. Finally, in (66), Sullivan’s method of quasiconformal conjugacy was adapted to show that a transcendental meromorphic function of finite type has no wandering domains. These four papers opened a new and fruitful area of research, made even more accessible by the excellent survey article [3], which appeared soon after. One of the differences between the iteration of entire functions and mero- morphic functions is the number of completely invariant components of the Fatou set that can occur. In (24) Noel proved that a transcendental entire function can have at most one completely invariant component of the Fatou set. In (64) Baker, Kotus and Lü proved that a transcendental meromorphic function of finite type can have at most two completely invariant Fatou com- ponents, and in this volume it is shown by Bergweiler and Eremenko that, in these circumstances, the Julia set must be a Jordan curve. (An example of a function with these properties is f(z) = tan z.) It is an open question whether a general transcendental meromorphic function can have at most two completely invariant Fatou components. Fatou-Julia theory can be developed in many further directions. For a tran- scendental meromorphic function f, the iterates fn need not be meromorphic. It is desirable, however, to have a closed system of iterates, so that we can consider, for example, the Fatou set of fn , for n ≥ 2. To obtain such a system, Noel’s student Herring [12], and independently Bolsch [5], developed Fatou-Julia theory for functions, such as f(z) = etan z , which are meromor- phic outside certain compact totally disconnected subsets of Ĉ. Much of this theory, and its subsequent developments, is expounded in Noel’s last papers (75), (77), (78) and (79). This volume also contains papers that, while not explicitly about complex dynamics, are on closely related topics. The paper by Hayman and Hinkka- nen is concerned with the growth of meromorphic functions that belong to certain normal families, the paper by Beardon and Minda classifies conformal automorphisms of finitely connected regions of the plane, and the paper by Szekeres is on possible connections between ‘regular growth’ and Abel’s func- tional equation, a topic in which Noel had a great interest. Finally, the paper by Bullett and Freiburger is on the theory of holomorphic correspondences, a generalisation of complex dynamics. Here they investigate, for the first time, holomorphic correspondences that involve transcendental entire functions.
- 19. INTRODUCTION xv Publications of I. N. Baker (1) The iteration of entire transcendental functions and the solution of the functional equation f(f(z)) = F(z), Math. Ann. 129 (1955) 174–180. (2) Zusammensetzungen ganzer Funktionen, Math. Z. 69 (1958) 121–163. (3) Fixpoints and iterates of entire functions, Math. Z. 71 (1959) 146– 153. (4) Solutions of the functional equation (f(x))2 − f(x2 ) = h(x), Canad. Math. Bull. 3 (1960) 113–120. (5) Some entire functions with fixpoints of every order, J. Austral. Math. Soc. 1 (1959/61) 203–209. (6) The existence of fixpoints of entire functions, Math. Z. 73 (1960) 280–284. (7) Permutable entire functions, Math. Z. 79 (1962) 243–249. (8) Permutable power series and regular iteration, J. Austral. Math. Soc. 2 (1961/62) 265–294. (9) Multiply-connected domains of normality in iteration theory, Math. Z. 81 (1963) 206–214. (10) Length of a graph, Solution in Amer. Math. Monthly 71 (1964) 217– 218. (11) Partition of a domain, Solution in Amer. Math. Monthly 71 (1964) 219–220. (12) Fractional iteration near a fixpoint of multiplier 1, J. Austral. Math. Soc. 4 (1964) 143–148. (13) Fixpoints of polynomials and rational functions, J. London Math. Soc. 39 (1964) 615–622. (14) Entire functions with linearly distributed values, Math. Z. 86 (1964) 263–267. (15) Sets of non-normality in iteration theory, J. London Math. Soc. 40 (1965) 499–502. (16) The distribution of fixpoints of entire functions, Proc. London Math. Soc. 16 (1966) 493–506. (17) On a class of meromorphic functions, Proc. Amer. Math. Soc. 17 (1966) 819–822. (18) On some results of A. Rényi and C. Rényi concerning periodic entire functions, Acta Sci. Math. (Szeged) 27 (1966) 197–200. (19) A series associated with the logarithmic function, J. London Math. Soc. 42 (1967) 336–338. (20) Non-embeddable functions with a fixpoint of multiplier 1, Math. Z. 99 (1967) 377–384. (21) (with F. GROSS) On factorizing entire functions, Proc. London Math. Soc. 18 (1968) 69–76. (22) Repulsive fixpoints of entire functions, Math. Z. 104 (1968) 252–256.
- 20. xvi INTRODUCTION (23) (with F. GROSS) Further results on factorization of entire functions, Entire Functions and Related Parts of Analysis (Proc. Symp. Pure Math. La Jolla, Calif., 1996) (Amer. Math. Soc., 1968) 30–35. (24) Completely invariant domains of entire functions, Mathematical Es- says Dedicated to A. J. Macintyre (ed. H. Shankar, Ohio Univ. Press. 1970) 33–35. (25) Limit functions and sets of non-normality in iteration theory, Ann. Acad. Sci. Fenn. Ser. A I Math. 467 (1970) 11 pp. (26) The value distribution of composite entire functions, Acta. Sci. Math. (Szeged) 32 (1971) 87–90. (27) (with L. S. O. LIVERPOOL) Picard sets for entire functions, Math. Z. 126 (1972) 230–238. (28) (with L. S. O. LIVERPOOL) Further results on Picard sets of entire functions, Proc. London Math. Soc. 26 (1973) 82–98. (29) Linear Picard sets for entire functions, Math. Nachr. 64 (1974) 263– 276. (30) (with J. A. DEDDENS and J. L. ULLMAN) A theorem on entire functions with applications to Toeplitz operators, Duke Math. J. 41 (1974) 739–745. (31) (with E. MUES) Zur Faktorisierung endlicher Blaschkeproductke, Arch. Math. (Basel) 26 (1975) 388–390. (32) The domains of normality of an entire function, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975) 277–283. (33) An entire function which has wandering domains, J. Austral. Math. Soc. Ser. A 22 (1976) 173–176. (34) Analytic mappings between two ultrahyperelliptic surfaces, Aequa- tiones Math. 14 (1976) 461–472. (35) (with C. C. YANG) An infinite order periodic entire function which is prime, Complex Analysis, Lecture Notes in Math. Vol 599 (Springer, 1977) 7–10. (36) (with L. S. O. LIVERPOOL) Sylvester series and normal families, Solution in Amer. Math. Monthly 85 (1978) 290–291. (37) (with L. S. O. LIVERPOOL) The value distribution of entire functions of order at most one, Acta Sci. Math. (Szeged) 41 (1979) 3–14. (38) (with Ch. POMMERENKE) On the iteration of analytic functions in a halfplane II, J. London Math. Soc. 20 (1979) 255–258. (39) Condition for a composite of polynomials, Solution in Amer. Math. Monthly 87 (1980) 228. (40) Entire functions with two linearly distributed values, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980) 381–386. (41) The iteration of polynomials and transcendental entire functions, J. Austral. Math. Soc. Ser. A 30 (1980/81) 483–495.
- 21. INTRODUCTION xvii (42) (with J. M. ANDERSON and J. G. CLUNIE) The distribution of values of certain entire and meromorphic functions, Math. Z. 178 (1981) 509–525. (43) Entire functions whose a-points lie on systems of lines, Factorization theory of meromorphic functions, Lecture Notes in Pure and Appl. Math. 78 (ed. C. C. Yang, Marcel Dekker, 1982) 1–18. (44) Complex function theory: a sequence of entire functions converging pointwise, James Cook Math. Notes, Townsville, Qld, Australia, Issue 29, Vol. 3 (August 1982) 3112–3114. (45) (with Z. RUBINSTEIN) Simultaneous iteration by entire or rational functions and their inverses, J. Austral. Math. Soc. Ser. A 34 (1983) 364–367. (46) (with P. J. RIPPON) Convergence of infinite exponentials, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983) 179–186. (47) (with L. S. O. LIVERPOOL) The entire solutions of a polynomial difference equation, Aequationes Math. 27 (1984) 97–113. (48) Composition of polynomials, Solution in Amer. Math. Monthly 91 (1984) 317. (49) (with P. J. RIPPON) Iteration of exponential functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984) 49–77. (50) Wandering domains in the iteration of entire functions, Proc. London Math. Soc. 49 (1984) 563–576. (51) (with P. J. RIPPON) A note on infinite exponentials, Fibonacci Quart. 23 (1985) 106–112. (52) (with P. J. RIPPON) A note on complex iteration, Amer. Math. Monthly 92 (1985) 501–504. (53) Some entire functions with multiply-connected wandering domains, Ergodic Theory Dynam. Systems 5 (1985) 163–169. (54) Wandering domains for maps of the punctured plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987) 191–198. (55) (with A. EREMENKO), A problem on Julia sets, Ann. Acad. Sci. Fenn. Ser. A I Math. 12 (1987) 229–236. (56) Iteration of entire functions: an introductory survey, Proceeding of the Symposium on Complex Analysis, 21–22 May 1987, Xian, China, Lectures on Complex Analysis (World Sci. Publishing, 1988) 1–17. (57) Infinite limits in the iteration of entire functions, Ergodic Theory Dy- nam. Systems 8 (1988) 503–507. (58) (with P. BHATTACHARRYA) On a class of non-embeddable entire functions, J. Ramanujan Math. Soc. 3 (1988) 151–159. (59) (with P. J. RIPPON) Iterating exponential functions with cyclic ex- ponents, Math. Proc. Cambridge Phil. Soc. 105 (1989) 357–375.
- 22. xviii INTRODUCTION (60) (with P. J. RIPPON) Towers of exponents and other composite maps, Complex Variables Theory Appl. , Volume in honour of Albert Edrei and Wolfgang Fuchs, 12 (1989) 181–200. (61) (with P. J. RIPPON) On compositions of analytic self-mappings of a convex domain, Arch. Math. (Basel) 55 (1990) 380–386. (62) (with J. KOTUS and LÜ YINIAN) Iterates of meromorphic functions II: Examples of wandering domains, J. London Math. Soc. 42 (1990) 267–278. (63) (with J. KOTUS and LÜ YINIAN) Iterates of meromorphic functions I, Ergodic Theory Dynam. Systems 11 (1991) 241–248. (64) (with J. KOTUS and LÜ YINIAN) Iterates of meromorphic functions III: Preperiodic domains, Ergodic Theory Dynam. Systems 11 (1991) 603–618. (65) (with J. WEINREICH) Boundaries which arise in the dynamics of entire functions, Analyse complexe (Bucharest, 1989), Rev. Roumaine Math. Pures Appl. 36 (1991) 413–420. (66) (with J. KOTUS and LÜ YINIAN) Iterates of meromorphic functions IV: Critically finite functions, Results Math. 22 (1992) 651–656. (67) (with R. N. MAALOUF) Convergence of a modified iteration process, Computational Methods and Function Theory, 1994 (Penang) (ed. S. Ruscheweyh, World Sci. Publishing, 1995) 49–55. (68) (with A. P. SINGH) Wandering domains in the iteration of composi- tions of entire functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995) 149–153. (69) (with A. P. SINGH ) A note on differential polynomials, Bull. Calcutta Math. Soc. 87 (1995) 63–66. (70) (with G. M. STALLARD) Error estimates in a calculation of Ruelle, Complex Variables Theory Appl. 29 (1996) 141–159. (71) On factorizing meromorphic functions, Aequationes Math. 54 (1997) 87–101. (72) (with P. DOMÍNGUEZ) Analytic self-maps of the punctured plane, Complex Variables Theory Appl. 37 (1998) 67–91. (73) (with P. DOMÍNGUEZ) Boundaries of unbounded Fatou components of entire functions, Ann. Acad. Sci. Fenn. Math. 24 (1999) 437–464. (74) (with P. DOMÍNGUEZ) Some connectedness properties of Julia sets, Complex Variables Theory Appl. 41 (2000) 371–389. (75) (with P. DOMÍNGUEZ) Residual Julia sets, J. Anal. 8 (2000) 121– 137. (76) Dynamics of slowly growing entire functions, Bull. Austral. Math. Soc. 63 (2001) 367–377. (77) (with P. DOMÍNGUEZ and M. HERRING) Dynamics of functions meromorphic outside a small set, Ergodic Theory Dynam. Systems 21 (2001) 647–672.
- 23. INTRODUCTION xix (78) Limit functions in wandering domains of meromorphic functions, Ann. Acad. Sci. Fenn. Math. 27 (2002) 499–505. (79) (with P. DOMÍNGUEZ and M. HERRING) Functions meromorphic outside a small set: completely invariant domains, Complex Variables Theory Appl. 49 (2004) 95–100. References [1] D. Bargmann, Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems 19 (1999) 553–558. [2] W. Bergweiler, Periodic points of entire functions: proof of a conjecture of Baker, Complex Variables Theory Appl. 17 (1991) 57–72. [3] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (1993) 151–188. [4] F. Berteloot and J. Duval, Une démonstration directe de la densité de cycles répulsif dans l’ensemble de Julia, Complex analysis and geometry (Paris 1997) , Progr. Math. 188 (Birkhäuser, Basel, 2000) 221–222. [5] A. Bolsch, Repulsive periodic points of meromorphic functions, Complex Variables Theory Appl. 31 (1996) 75–79. [6] A. E. Eremenko and M. Yu. Lyubich, Examples of entire functions with pathological dynamics, J. London Math. Soc. 36 (1987) 458–468. [7] A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble) 42 (1992) 989–1020. [8] P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. France 47 (1919) 161–271; 48 (1920) 33–94, 208–314. [9] P. Fatou, Sur l’itération des fonctions transcendantes entières, Acta Math. 47 (1926) 337–370. [10] L. Goldberg and L. Keen, A finiteness theorem for a dynamical class of entire functions, Ergodic Theory Dynam. Systems 6 (1986) 183–192. [11] W. K. Hayman, Meromorphic functions (Clarendon Press, Oxford, 1964). [12] M. Herring, An extension of the Julia–Fatou theory of iteration, Ph.D. thesis, Univer- sity of London (1994). [13] G. Julia, Mémoire sur l’itération des fonctions rationelles, J. Math. Pure Appl. 1 (1918) 47–245. [14] S. Lattès, Sur l’itération des substitutions rationelles et les fonctions de Poincaré, C.R. Acad. Sci. Paris Sér. I Math. 166 (1918) 26–28 (Errata: p. 88). [15] M. Misiurewicz, On iterates of ez , Ergodic Theory Dynam. Systems 1 (1981) 103–106. [16] D. Sullivan, Quasiconformal homeomorphisms and dynamics I, Ann. Math. 122 (2) (1985) 401–418. [17] P. J. Rippon, Obituary: Irvine Noel Baker 1932–2001, Bull. London Math. Soc. 37 (2005) 301–315. [18] W. Schwick, Repelling periodic points in the Julia set, Bull. London Math. Soc. 29 (1997) 314–316. [19] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Scient. Ec. Norm. Sup. (4) 20 (1987) 1–29.
- 25. ITERATION OF INNER FUNCTIONS AND BOUNDARIES OF COMPONENTS OF THE FATOU SET DETLEF BARGMANN Abstract. Let D be an unbounded invariant component of the Fatou set of a transcendental entire function f. Let φ : D → D be a Riemann map. Then the set Θ := {θ ∈ ∂D : limr→1 φ(rθ) = ∞} is closely related to the Julia set of the corresponding inner function g := φ−1 ◦ f ◦ φ. In the first part of the paper we further develop the theory of Julia sets of inner functions and the dynamical behaviour on their Fatou sets. In the second part we apply these results to iteration of entire functions by using the above relation and obtain some new results about the boundaries of components of the Fatou set of an entire function. 1. Introduction Dynamics of inner functions have turned out to be a very useful tool to study the boundary structure of unbounded invariant components of the Fa- tou set of a transcendental entire function. The key method in this area has been developed by I.N. Baker and P. Domı́nguez [3]. We recall some of their techniques and results in Subsection 1.1 and then state the aims and results of this paper in Subsection 1.2 1.1. The method of Baker-Domı́nguez. Let f be a transcendental entire function, with Fatou set F(f) and Julia set J (f); see [7] for background information on these concepts. Suppose that D is an unbounded invariant component of F(f). Then D is simply connected [2, Theorem 1], which implies that there exists a biholomorphic (Riemann) map φ : D → D. Then g := φ−1 ◦ f ◦ φ is an inner function, i.e. a holomorphic self-map of the unit disk such that lim r→1 |g(r exp(2πiα))| = 1, for almost every α ∈ [0, 1]. According to Fatou’s theorem [18, p. 139] this implies that limr→1 g(r exp(2πiα)) exists and is contained in ∂D, for almost every α ∈ [0, 1]. Keywords: inner function, Julia set, Fatou set, boundary, radial limit, Baker domain. AMS subject classification: 30D05, 58F08. 1
- 26. 2 DETLEF BARGMANN If f|D is a proper self-map of D, then g is a (finite) Blaschke product, i.e. there exist m, n ∈ N0, λ ∈ ∂D, and a1, . . . , an ∈ D {0} such that g(z) = λzm n Y j=1 z − aj 1 − ajz , for each z ∈ D. Since D is unbounded f|D need not be a proper self-map of D. In this case g has at least one singularity on the boundary of the unit disk. Definition 1.1. Let g be an inner function of D. A point ζ ∈ ∂D is called a singularity of g if g cannot be continued analytically to a neighbourhood of ζ. Denote the set of all singularities of g by sing(g). Throughout this paper we assume that an inner function is always contin- ued to ĈD by the reflection principle, where Ĉ denotes the complex sphere, and to ∂D sing(g) by analytic continuation. It follows from the theory of inner functions that the composition of two inner functions is again an inner function; see [3, Lemma 4]. In particular, the n-th iterate gn of an inner function g is an inner function. Now, the Julia set of an inner function can be defined in the following way. Definition 1.2. Let g be an inner function of the unit disk D. The Fa- tou set F(g) of g is the set of all points z ∈ Ĉ for which there is an open neighbourhood U ⊂ Ĉ of z such that U ∩ sing(gn ) = ∅, for each n ∈ N, and {gn |U : n ∈ N} is normal. The Julia set J (g) of g is the complement of F(g) in Ĉ. Remark 1.3. It follows from Montel’s theorem that J (g) ⊂ ∂D. Moreover, for the case of a finite Blaschke product this definition coincides with the usual definition of the Julia set of a rational function. Baker and Domı́nguez initiated the study of Julia sets of inner functions by proving the following result [3, Lemma 8]. Theorem 1.4 (Baker-Domı́nguez). Let g be an inner function of the unit disk D. Then the following properties hold. (1) g(F(g)) ⊂ F(g). (2) If g is non-Möbius, then J (g) is a perfect set. (3) If g is non-rational, then J (g) = S n∈N sing(gn). The main tool in the proof of Theorem 1.4 is the the following lemma on inner functions [3, Lemma 5], which we use later. For the definition of a Stolz angle we refer the reader to [18, p. 6]. Lemma 1.5 (Baker-Domı́nguez). Let g be an inner function of the unit disk. Suppose that ζ ∈ sing(g). Then, for each θ ∈ ∂D and each neighbourhood U of ζ, there exist η ∈ U {ζ} and a path γ : [0, 1) → D such that limt→1 γ(t) = η and g(γ(t)) → θ in a Stolz angle as t → 1.
- 27. ITERATION OF INNER FUNCTIONS 3 Following Baker and Domı́nguez [3], we relate the boundary behaviour of the Riemann map φ to the dynamical behaviour of the corresponding inner function g. This process involves the sets Ξ := {ζ ∈ ∂D : ∞ ∈ C(φ; ζ)}, and Θ := {θ ∈ ∂D : lim r→1 φ(rθ) = ∞}. Here C(φ; ζ) denotes the cluster set of φ at ζ, i.e. the set of all values w ∈ Ĉ for which there is sequence (zn)n∈N in D such that zn → ζ and φ(zn) → w as n → ∞. Since D is unbounded the set Ξ is always non-empty. In general, it is not known whether the set Θ is always non-empty. However, for the case when D is a Baker domain, i.e. fn |D → ∞ locally uniformly, it is easy to see that Θ 6= ∅. Throughout this paper the sets Ξ and Θ will relate to a Riemann mapping φ : D → D of the invariant Fatou component D of the function f under consideration. There is a close connection between Ξ and J (g). Lemma 1.6. If f|D is not an automorphism of D, then J (g) ⊂ Ξ. Proof. First, it is easy to see that Ξ is closed. Case 1. Suppose that g is rational. Then g is a finite Blaschke product and it is easy to see that Ξ is backward invariant under g. Since g is locally injective on ∂D (see Remark 2.19) we conclude that Ξ is an infinite set. Hence Ξ is a closed, backward invariant set which contains at least three points. This implies that Ξ is a superset of J (g). Case 2. Suppose that g is not rational. By Theorem 1.4 we need only show that the singularities of the iterates of g are contained in Ξ. Let n ∈ N and ζ be a singularity of gn . Then C(gn ; ζ) = D (see for instance [13, Theorem 5.4]), which implies that ∞ ∈ D ⊂ C(φ ◦ gn ; ζ) = C(fn ◦ φ; ζ). Thus we conclude that ∞ ∈ C(φ; ζ). 2 Using Lemma 1.5, Baker and Domı́nguez obtained a similar result for the set Θ; see [3, Lemma 13]. Lemma 1.7 (Baker-Domı́nguez). Suppose that f|D is not an automorphism of D and Θ 6= ∅. Then J (g) ⊂ Θ. Hence the Julia set of the corresponding inner function g is a lower bound for the size of the set Ξ and, provided that Θ 6= ∅, for the size of the set Θ. This provides a strategy to show that the sets Ξ and Θ are equal to the unit circle by showing that the Julia sets of the corresponding inner functions are the unit circle. Baker and Domı́nguez gave two cases when J (g) = ∂D; see [3, Lemma 9 and Lemma 10].
- 28. 4 DETLEF BARGMANN Theorem 1.8 (Baker-Domı́nguez). Let g be a non-Möbius inner function of D with a fixed point p ∈ D. Then J (g) = ∂D. Theorem 1.9 (Baker-Domı́nguez). Let g be an inner function of D. Suppose that there exists p ∈ ∂D such that, for each z ∈ D, gn (z) → p in an arbitrarily small Stolz angle as n → ∞. Then g is non-Möbius and J (g) = ∂D. The assumptions of Theorem 1.8 are satisfied for the corresponding inner function g when D is an attracting domain of f. Theorem 1.9 can be applied to the corresponding inner function g when D is a parabolic domain of f. Taken together, these lead to the following result; see [4, Theorem 1] and [3, Theorem 1.1]. Theorem 1.10 (Baker-Weinreich, Baker-Domı́nguez). Suppose that D is an attracting domain, a parabolic domain, or a Siegel disk of f. Then Ξ = ∂D and Θ 6= ∅ ⇒ Θ = ∂D. This result does not carry over in general to Baker domains, since there are examples of Baker domains (due to Baker-Weinreich and Bergweiler, see Examples 3.4 and 3.5 in Subsection 3.1) whose boundaries are Jordan curves. On the other hand, Baker and Weinreich proved [4, Theorem 4] that if D is a Baker domain whose boundary is a Jordan curve, then f|D has to be univalent. Thus it is natural to ask whether Theorem 1.10 holds for those Baker domains for which f|D is not univalent. In this case, Baker and Domı́nguez proved that Θ contains a perfect set and hence is infinite; see [3, Theorem 1.2]. 1.2. Aims and results of this paper. The aim of this paper is twofold. In Section 2 we further develop the theory of Julia sets of inner functions, independently of the application to iteration of entire functions. Then, in Section 3, we apply these results to iteration of entire functions. Here, we use the method of Baker-Domı́nguez described in Subsection 1.1 and further extend it. In Section 2, the main results are in Subsections 2.3 and 2.4. In Subsec- tion 2.3 we prove the following theorem, which is a generalization of Theo- rems 1.8 and 1.9. For a hyperbolic domain G in the complex sphere Ĉ, λG always denotes the hyperbolic metric on G. Theorem 2.24 Let g be an inner function such that λD(gn (z), gn+1 (z)) → 0 as n → ∞ for some z ∈ D. Then J (g) = ∂D. Theorem 2.24 will be an easy consequence of two more general theorems which give necessary and sufficient conditions for an inner function to be eventually conjugated to a certain Möbius transformation on the Fatou set and which also classify the possible eventual conjugacies which can arise. See Subsection 2.1 for the meaning of ‘eventual conjugacy’.
- 29. ITERATION OF INNER FUNCTIONS 5 The different types of components of F(g) ∩ ∂D will be introduced and classified in Subsection 2.2. In Subsection 2.4 we prove that the Julia set of an inner function coin- cides with the closure of the set of repelling periodic points (with a suitable definition of repelling periodic point, see Definition 2.30). Analogous results are known for rational and entire functions and it is an interesting fact that inner functions have this property, too. Theorem 2.34 Let g be a non-Möbius inner function. Then J (g) is the closure of the set of the repelling periodic points of g. At the end of Section 2, we give some examples of inner functions; in particular, we show that all the possible types of eventual conjugacy can occur. In Section 3, we apply the results of Section 2 to the iteration of entire functions. For instance, using Lemma 1.7 and our Theorem 2.24 we can prove the following generalization of Theorem 1.10. Theorem 3.2 Let f be a transcendental entire function. Suppose that D is an unbounded invariant component of the Fatou set of f such that λD(fn (z), fn+1 (z)) → 0 as n → ∞, for some z ∈ D. If Θ 6= ∅, then Θ = ∂D. We can use Theorem 3.2 to extend Theorem 1.10 to a certain class of Baker domains. Here we use the symbol ∼ to indicate an eventual conjugacy. Theorem 3.1 Let f be a transcendental entire function. Suppose that D is a Baker domain of f such that f|D ∼ idC + 1. Then Θ = ∂D. Moreover, we see in Subsection 3.1 that, for a whole class of examples, the set Θ is dense in ∂D whenever f is not univalent on the Baker domain D; see Lemma 3.3. In Subsection 3.2 we improve Lemma 1.7, at least for the case when D is a completely invariant component of the Fatou set. Theorem 3.8 Let f be a transcendental entire function. Suppose that D is a completely invariant component of the Fatou set of f. Let φ : D → D be a Riemann map and let g := φ−1 ◦ f ◦ φ be the corresponding inner function. If Θ 6= ∅, then J (g) is equal to the set of accumulation points of Θ. As a consequence of this result, we shall be able to prove the following results about boundaries of components of the Fatou set; see Subsection 3.3.
- 30. 6 DETLEF BARGMANN Theorem 3.11 Let f be a transcendental entire function. Suppose that D is a completely invariant component of the Fatou set of f. Let G ⊂ C be a bounded Jordan domain such that G ∩ J (f) 6= ∅. Then ∂G ∩ D has infinitely many components. Theorem 3.12 Let f be a transcendental entire function. Suppose that there exists an unbounded component of the Fatou set of f. Let G ⊂ C be a bounded Jordan domain such that G ∩ J (f) 6= ∅. Then ∂G ∩ F(f) has infinitely many components. Theorem 3.14 Let f be an entire function. Let D be a component of the Fatou set of f. Suppose that at least one of the following conditions is satisfied: (1) f is transcendental and there exists an unbounded component of the Fatou set of f, or (2) S n∈N fn (D) is bounded. Let φ : D → D be a Riemann map. Let Acc(D) be the set of finite accessible boundary points of D, and let Z be the set of all ζ ∈ ∂D such that φ(ζ) := limr→1 φ(rζ) exists and is finite. Then the map Z → Acc(D), ζ 7→ φ(ζ) is a bijection. Corollary 3.15 Let f be an entire function. Suppose that D is a Siegel disk for f. Then (1) There is no periodic point of f in ∂D which is an accessible boundary point of D. (2) f is univalent on the set of finite accessible boundary points of D. Theorem 3.16 Let f be an entire function. Let E be a finite set of com- ponents of the Fatou set of f. Suppose that at least one of the following conditions is satisfied: (1) f is transcendental and there exists an unbounded component of the Fatou set of f, or (2) S n∈N fn (D) is bounded, for each D ∈ E. Then there are at most card(E) − 1 points in C which are common accessible boundary points of at least two components in E. Acknowledgement. I would like to thank the late Professor I.N. Baker and Professor W. Bergweiler for their support and helpful discussions. 2. Iteration of inner functions 2.1. Holomorphic self-maps of hyperbolic domains. In this subsection we recall some facts about the dynamical behaviour of a holomorphic self-map
- 31. ITERATION OF INNER FUNCTIONS 7 of the unit disk or, more generally, of a hyperbolic domain in the complex sphere. We start with an old theorem of A. Denjoy [15] and J. Wolff [22]; see also [12, p. 79] or [21, p. 43]. For the notion of the angular limit we refer the reader to [18, p. 6]. Theorem 2.1 (Denjoy-Wolff). Let h be a non-Möbius holomorphic self-map of D. Then there is a point p ∈ D such that hn → p locally uniformly on D. Moreover, if p ∈ ∂D, then h has the angular limit p at p. The point p referred to in this theorem is often called the Denjoy-Wolff point of h. It may appear that the dynamical behaviour of a holomorphic self-map depends only on the question whether its Denjoy-Wolff point p is inside the disk or on its boundary. But, in fact, the case p ∈ ∂D can be further subdivided. Here we make use of a classification due to C. Cowen [14] who has shown that a holomorphic self-map of the unit disk is eventually conjugated to a certain Möbius transformation. Roughly speaking, eventually conjugated means that the function is semi-conjugated to a Möbius transfor- mation and, starting at an arbitrary point and iterating, one eventually lands in a simply connected region where the function is even conjugated to this Möbius transformation. More precisely, we have the following definition. Definition 2.2. Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ. Let T be a biholomorphic self-map of a simply connected domain Ω ⊂ C. Then we say that h ∼ T (h is eventually conjugated to T) if there exist a holomorphic function Φ : G → Ω and a simply connected domain V ⊂ G such that the following conditions are satisfied: (1) Φ ◦ h = T ◦ Φ. (2) Φ is univalent on V . (3) V is a fundamental set for h on G, i.e. h(V ) ⊂ V and ∀ z ∈ G ∃ n ∈ N : hn (z) ∈ V. (4) Φ(V ) is a fundamental set for T on Ω. In this case, (Ω, T, Φ, V ) is called an eventual conjugacy of h on G. Eventual conjugacies are unique in the following sense; see [14, p. 79-80]. Lemma 2.3 (Cowen). Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ. Let (Ω1, T1, Φ1, V1) be an eventual conjugacy of h on G. (1) Let τ : Ω1 → C be an injective holomorphic function. Then (τ(Ω1), τ ◦ T1 ◦ τ−1 , τ ◦ Φ1, V1) is an eventual conjugacy of h on G. (2) Let (Ω2, T2, Φ2, V2) be another eventual conjugacy of h on G. Then there exists a component W of V1 ∩ V2 such that W is a fundamental set for h on G and, for each j ∈ {1, 2}, Φj(W) is a fundamental set for Tj on Ωj. Moreover, there exists a biholomorphic map τ : Ω1 → Ω2 such that Φ2 = τ ◦ Φ1 and T2 = τ ◦ T1 ◦ τ−1 .
- 32. 8 DETLEF BARGMANN Cowen’s result can be stated as follows. Here, H := {z ∈ C : Im(z) 0}. Theorem 2.4 (Cowen). Let h be a holomorphic self-map of the unit disk D without a fixed point. Then exactly one of the following statements holds. (1) h ∼ idC + 1. (2) There exists exactly one σ ∈ {−1, 1} such that h ∼ idH + σ. (3) There exists exactly one λ 1 such that h ∼ λidH. For the proof of Theorem 2.4, see [14, Theorem 3.2] and the following remark. Remark 2.5. By applying Lemma 2.3 it is easy to see that, for each σ ∈ {−1, 1}, h ∼ idH + σ ⇐⇒ h ∼ idσH + 1 ⇐⇒ h ∼ id−iH − iσ and, for each λ 1, h ∼ λidH ⇐⇒ h ∼ λid−H ⇐⇒ h ∼ λid−iH. H. König [17] has given geometrical conditions to determine which Möbius transformation the function h is eventually conjugated to. Expressing these conditions in terms of the hyperbolic metric we obtain the following lemma. Recall that we denote the hyperbolic metric on a hyperbolic domain G ⊂ Ĉ by λG. Lemma 2.6. Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ without a fixed point. For each z ∈ G, let ρ(z) := inf n∈N λG(hn (z), hn+1 (z)) = lim n→∞ λG(hn (z), hn+1 (z)). Then we have that (1) h ∼ idC + 1 ⇒ ρ = 0, (2) h ∼ id±H + 1 ⇒ ρ 0 and inf ρ(G) = 0, (3) ∃ λ 1 : h ∼ λidH ⇒ inf ρ(G) 0. Remark 2.7. If additionally G is simply connected, then Cowen’s result im- plies that all these implications are equivalences. In general, a holomorphic self-map of a hyperbolic domain need not be semi-conjugated to a Möbius transformation at all. To prove Lemma 2.6 we make use of the following lemma. Lemma 2.8. Let Ω ∈ {C, H, −H} and let T := idΩ + 1. Let W ⊂ Ω be a fundamental set for T on Ω. Let r 1 and let w ∈ Ω be the center of an open disk Q with radius r such that Q ⊂ Ω. Then lim n→∞ λW (w + n, w + n + 1) ≤ 1 2 log µ r + 1 r − 1 ¶ .
- 33. ITERATION OF INNER FUNCTIONS 9 Proof. Since W is a fundamental set for idΩ + 1 on Ω there exists n ∈ N such that Q + n ⊂ W. By the Schwarz-Pick lemma we have that λW (w + n, w + n + 1) ≤ λQ+n(w + n, w + n + 1) = 1 2 log µ 1 + 2 r − 1 ¶ . 2 Proof of Lemma 2.6. Let (Ω, T, Φ, V ) be an eventual conjugacy of h on G, where Ω ∈ {C, H, −H}. Let W := Φ(V ). For each z ∈ G and n ∈ N such that hn (z) ∈ V , the Schwarz-Pick lemma implies that (∗) λΩ(Φ(z), T(Φ(z))) = λΩ(Tn (Φ(z), Tn+1 (Φ(z))) = λΩ(Φ(hn (z)), Φ(hn+1 (z))) ≤ λG(hn (z), hn+1 (z)) ≤ λV (hn (z), hn+1 (z)) = λW (Φ(hn (z)), Φ(hn+1 (z))) = λW (Tn (Φ(z)), Tn+1 (Φ(z))), where the first inequality makes sense only for the case when Ω ∈ {H, −H}. To prove (3) we observe that, for each λ 1 and w ∈ H, λH(λw, w) = log 1 + (λw − w)/(λw − w) 1 − (λw − w)/(λw − w) ≥ log λ, which together with (∗) implies that inf ρ(G) 0 if h ∼ λidH. To prove (2) suppose that there exists σ ∈ {−1, 1} such that T = idσH +1. Then, for each w ∈ σH, we have that λσH(w + 1, w) = log 1 + |σ/(σ + 2iIm(w))| 1 − |σ/(σ + 2iIm(w))| ≥ log à 1 + 1 |Im(w)| ! , which together with (∗) implies that ρ 0. Because of (∗) it remains to show that inf w∈W inf n∈N λW (w + n, w + n + 1) = 0. This is an easy consequence of Lemma 2.8 because σH contains an open disk with an arbitrarily large radius. To prove (1) suppose that T = idC + 1. Because of (∗) it remains to show that, for each w ∈ C, inf n∈N λW (w + n, w + n + 1) = 0. This is an easy consequence of Lemma 2.8 because each w ∈ C is the center of an open disk in C with arbitrarily large radius. 2 The case when the Denjoy-Wolff point of a holomorphic self-map of D is inside the unit disk also leads to an eventual conjugacy. More generally, we have the following result. Although this lemma might be folklore, for the sake of completeness we give a short proof.
- 34. 10 DETLEF BARGMANN Lemma 2.9. Let h be a holomorphic self-map of a hyperbolic domain G ⊂ Ĉ such that h is not an automorphism of G and there is a fixed point p of h in G such that λ := h0 (p) 6= 0. Then |λ| 1 and h ∼ λidC. Proof. By Montel’s theorem, the family {hn : n ∈ N} is normal. Since h is not an automorphism of G we conclude that no subsequence of (hn )n∈N con- verges to a non-constant limit function (see for instance [6, Theorem 7.2.4]). Hence hn → p locally uniformly on G, which implies that |λ| 1. Hence there is an open and connected neighbourhood V of p in G, a number r 0, and a biholomorphic map φ : V → D(0, r) such that h(V ) ⊂ V , φ(p) = 0, φ0 (p) = 1 and φ ◦ h|V = λφ. It is easy to see that Φ(z) := 1 λn φ(hn (z)) if hn (z) ∈ V is a well-defined holomorphic function on G such that (C, λidC, Φ, V ) forms an eventual conjugacy of h on G. 2 Moreover, we make use of the following theorems due to P. Bonfert [11, Theorem 5.7 and Theorem 6.1]. Theorem 2.10 (Bonfert). Let h be a holomorphic self-map of a hyperbolic domain G ⊂ C without a fixed point, and without an isolated boundary fixed point, i.e. there is no isolated boundary point a ∈ ∂G such that h extends holomorphically to a and fixes a. Suppose that λG(hn (z), hn+1 (z)) → 0 as n → ∞ for some z ∈ G. Let z0 ∈ G. Define φn : G → C, z 7→ hn (z) − hn (z0) hn+1(z0) − hn(z0) . Then the sequence (φn)n∈N converges locally uniformly in G to a holomorphic function φ : G → C such that φ(h(z)) = φ(z) + 1 for all z ∈ G. Theorem 2.11 (Bonfert). Let T be a Möbius transformation and G ⊂ C be a hyperbolic domain such that T(G) ⊂ G, T(∞) = ∞, and T has no fixed point in G. Then λG(Tn (z), Tn+1 (z)) → 0 as n → ∞ for (any) z ∈ G if and only if [ n∈N T−n (G) = C or [ n∈N T−n (G) = C {b}, where b ∈ C G is a fixed point of T.
- 35. ITERATION OF INNER FUNCTIONS 11 2.2. The components of F(g) ∩ ∂D. We distinguish between the following types of components of F(g) ∩ ∂D. Definition 2.12. Let g be a non-Möbius inner function such that J (g) 6= ∂D. A component I of F(g) ∩ ∂D is called (1) absorbing, if, for each component J of F(g) ∩ ∂D, there exists n ∈ N such that gn (J) ⊂ I, (2) invariant, if g(I) ⊂ I, (3) eventually invariant, if there exists n ∈ N such that gn (I) is contained in an invariant component of F(g) ∩ ∂D, (4) wandering, if, for all m 6= n ∈ N, gm (I) and gn (I) are contained in different components of F(g) ∩ ∂D. As a consequence of Theorem 2.1 we can make the following observation. Lemma 2.13. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Then there is at most one invariant component of F(g) ∩ ∂D. Proof. Let p ∈ D be the Denjoy-Wolff point of g. By Theorem 1.8 we have that p ∈ ∂D. Since gn |F(g) → p we conclude that p ∈ I for each invariant component I of F(g) ∩ ∂D. Since J (g) is perfect there is at most one component J of F(g)∩∂D such that p ∈ J. Hence the conclusion follows. 2 This leads to the following dichotomy of the set of components of F(g)∩∂D. Lemma 2.14. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Then each component of F(g) ∩ ∂D is either eventually invariant or wander- ing. Moreover, each absorbing component of F(g) ∩ ∂D is invariant. Proof. It remains to show that each component I of F(g) ∩ ∂D for which there exists n ∈ N such that gn (I) ⊂ I is invariant. Let I be a component of F(g) ∩ ∂D and n ∈ N such that gn (I) ⊂ I. Let J be the component of F(g)∩∂D which contains g(I). Then I and J are both invariant components of F(gn ) ∩ ∂D which by Lemma 2.13 implies that I = J. 2 Remark 2.15. The converse of the second statement in Lemma 2.14 is not true. There may be an invariant component which is not absorbing (see Ex- ample 2.38 in Subsection 2.5). The next theorem gives some information about the mapping behaviour of an inner function g on a component of F(g) ∩ ∂D. Definition 2.16. Let ζ, θ ∈ ∂D such that ζ 6= θ. Let α ∈ (0, 2π) such that θ = ζ exp(iα). Then define (ζ, θ) := ζ exp(i(0, α)). Theorem 2.17. Let g be a non-Möbius inner function. Let I = (ζ1, ζ2) be a component of F(g) ∩ ∂D. Then
- 36. 12 DETLEF BARGMANN (1) g|I is injective. (2) ηj := limz→ζj g|I(z) exists, for each j ∈ {1, 2}, and is equal to the angular limit of g at ζj. Moreover, η1 6= η2, and g(I) = (η1, η2) or g(I) = (η2, η1). We make use of the following lemmata to prove Theorem 2.17. Lemma 2.18. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D be distinct and let g be holomorphic on I := (ζ, θ). Suppose that η := lim z→ζ g|I(z) and lim z→θ g|I(z) exist and are equal. Then ∂D {η} ⊂ g(I). Proof. The set Z := g(I) {η} is a non-empty open subset of ∂D {η}. Let w ∈ Z {η}. Then there exists a sequence (xn)n∈N in I such that limn→∞ g(xn) = w 6= η. We may assume that xn → x ∈ I. Since limz→ζ g|I(z) = η = limz→θ g|I(z) we conclude that x ∈ I and w ∈ Z. Since ∂D {η} is connected we have that Z = ∂D {η}. 2 Remark 2.19. As a consequence of Lemma 2.18 we obtain that an inner function g is locally injective at any point ζ ∈ ∂D where it is holomorphic. This can also be concluded from the Julia-Wolff lemma (see Theorem 2.29). Lemma 2.20. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D such that ζ 6= θ and g is holomorphic on I := (ζ, θ). Then either g((ζ, η)) = ∂D, for each η ∈ I, or limz→ζ g|I(z) exists. Proof. Suppose that there exists η ∈ I such that g((ζ, η)) 6= ∂D. From Lemma 2.18 we conclude that g is injective on (ζ, η). Hence g is strictly orientation preserving or strictly orientation reversing on (ζ, η). In both cases it is easy to see that limz→ζ g|I(z) exists. 2. The next result is an easy consequence of the Lehto-Virtanen theorem. Lemma 2.21. Let g be a non-Möbius inner function. Let ζ, θ ∈ ∂D such that ζ 6= θ and g is holomorphic on I := (ζ, θ). Suppose that η := limz→ζ g|I(z) exists. Then the angular limit of g at ζ exists and equals η. Proof. Choose a disk D which has its center at a point α ∈ I such that ζ ∈ ∂D, D∩∂D ⊂ I, and g(D∩∂D) ⊂ D(η; 1). Then D(−η; 1)∩∂D ⊂ Ĉg(D) which implies that g is a normal function on D. Thus from the Lehto-Virtanen theorem (see [18], p. 71) we have that there is a path γ : [0, 1) → D ∩ D such that γ(t) → ζ and γ(t) → η as t → 1. Again we apply the Lehto-Virtanen theorem and conclude that the angular limit of g at ζ exists and equals η. 2
- 37. ITERATION OF INNER FUNCTIONS 13 Proof of Theorem 2.17. Since g(I) ⊂ F(g) ∩ ∂D and J (g) 6= ∅ we conclude from Lemma 2.18 and Lemma 2.20 that g is injective on I, ηj := limz→ζj g|I(z) exists, for each j ∈ {1, 2}, η1 6= η2, and {η1, η2} ∩ g(I) = ∅. From this it can be easily deduced that g(I) = (η1, η2) or g(I) = (η2, η1). Moreover, it follows from Lemma 2.21 that the angular limit of g at ζj exists, for each j ∈ {1, 2}, and equals ηj. 2 2.3. Eventual conjugacies of inner functions. The following theorem gives necessary and sufficient conditions for an inner function to be eventually conjugated to a Möbius transformation on the whole Fatou set. Moreover, it restricts the eventual conjugacies that can occur. Theorem 2.22. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Then the following statements are equivalent. (1) There exist a simply connected domain Ω ⊂ C and a biholomorphic self-map T of Ω such that g|F(g) ∼ T. (2) There exists an absorbing component I of F(g) ∩ ∂D. (3) There is no wandering component of F(g) ∩ ∂D. Let p ∈ ∂D be the Denjoy-Wolff point of g. If one (and hence all) of these equivalent conditions (1), (2), (3) is satisfied, then exactly one of the following statements holds. (4) p ∈ F(g) and g|F(g) ∼ g0 (p)idC and g|D ∼ g0 (p)idH. (5) p ∈ J (g) and g|F(g) ∼ idC + 1 and g|D ∼ idH ± 1. (6) p ∈ J (g) and there exists λ 1 such that g|F(g) ∼ λidH and g|D ∼ λ2 idH. We see in Subsection 2.5 that all the listed cases (4),(5), and (6) may occur. For the case when g|F(g) ∼ idC + 1 we get two more equivalences. Theorem 2.23. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Let p ∈ ∂D be the Denjoy-Wolff point of g. Then the following conditions are equivalent: (1) g|F(g) ∼ idC + 1. (2) p ∈ J (g) and λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞ for some (all) z ∈ F(g). (3) g|D ∼ id±H +1 and there exists an absorbing component of F(g)∩∂D. (4) g|D ∼ id±H + 1 and there exists an invariant component of F(g) ∩ ∂D. As a consequence of Theorem 2.22 and 2.23 we obtain the following result which is a generalization of Theorems 1.8 and 1.9. Theorem 2.24. Let g be an inner function such that λD(gn (z), gn+1 (z)) → 0 as n → ∞ for some z ∈ D. Then J (g) = ∂D. Proof. Let z ∈ D such that λD(gn (z), gn+1 (z)) → 0 as n → ∞. Then g is not a Möbius transformation. Let p be the Denjoy-Wolff point of g. If p ∈ D,
- 38. 14 DETLEF BARGMANN then the conclusion follows from Theorem 1.8. Hence we may assume that p ∈ ∂D. From Lemma 2.6 we see that g|D ∼ idC + 1. Assume that p ∈ F(g). Then λ := g0 (p) ∈ (0, 1] and from Lemma 2.9 we conclude that g|F(g) ∼ λidC. By Theorem 2.22 this implies that g|D ∼ λidH. This is a contradiction to g|D ∼ idC + 1. Hence p ∈ J (g). Assume that F(g) ∩ ∂D 6= ∅. From the Schwarz-Pick lemma we conclude that λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞. By Theorem 2.23 this implies that g|D ∼ idH ± 1. This contradicts g|D ∼ idC + 1. Hence J (g) = ∂D. 2 This leads to the following theorem. Theorem 2.25. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Suppose that there exists an invariant component of F(g) ∩ ∂D which is not absorbing. Then there exists λ 1 such that g|D ∼ λidH. Proof. From Theorem 2.24 we know that g|D 6∼ idC+1. From Theorem 2.23 we conclude that g|D 6∼ idH ± 1. From Theorem 1.8 we see that g does not have a fixed point in D. Hence the conclusion follows from Theorem 2.4. 2 Remark. An example of a non-Möbius inner function which has an invari- ant but not absorbing component of F(g) ∩ ∂D is given in Subsection 2.5. The rest of this subsection is devoted to the proofs of Theorem 2.22 and 2.23. The main part of the proofs is contained in the following lemma. Lemma 2.26. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Let p ∈ ∂D be the Denjoy-Wolff point of g. Suppose that there exists an invariant component I of F(g) ∩ ∂D. Let G := D ∪ I ∪ (Ĉ D). Then exactly one of the following statements holds. (1) p ∈ F(g) and g|G ∼ g0 (p)idC and g|D ∼ g0 (p)idH. (2) p ∈ J (g) and g|G ∼ idC + 1 and g|D ∼ idH ± 1. (3) p ∈ J (g) and there exists λ 1 such that g|G ∼ λidH and g|D ∼ λ2 idH. Proof. By conjugating with an appropriate transformation we may assume that g is an inner function of the upper half plane H with Denjoy-Wolff point p = ∞ and that I is an invariant component of F(g)∩(R∪{∞}). Since J (g) is a perfect set the simply connected domain G := H ∪ I ∪ −H is hyperbolic. By Theorem 2.4 and Lemma 2.9 there exist (Ω, T) ∈ {(C, idC + 1)} ∪ {(−iH, id−iH + iσ) : σ ∈ {−1, 1}} ∪ {(−iH, λid−iH) : λ 1} ∪ {(C, g0 (p)idC}, a simply connected domain V ⊂ G, and a holomorphic function Φ : G → Ω such that (Ω, T, Φ, V ) forms an eventual conjugacy of g on G. Here (Ω, T) = (C, g0 (p)idC) if and only if ∞ = p ∈ I. By Lemma 2.13 this is the case if and only if p ∈ F(g).
- 39. ITERATION OF INNER FUNCTIONS 15 By the reflection principle {z : z ∈ V } is also a fundamental set for g on G and Ψ(z) := Φ(z) is holomorphic such that, for each z ∈ G, Ψ(g(z)) = Φ(g(z)) = Φ(g(z)) = T(Φ(z)) = S(Ψ(z)), where S(w) := T(w), for each w ∈ Ω. Thus we have that g|G ∼ T and g|G ∼ S. Assume that there exists σ ∈ {−1, 1} such that (Ω, T) = (−iH, id−iH +iσ). Then S = id−iH − iσ and hence g|G ∼ id−iH + i and g|G ∼ id−iH − i. This is a contradiction to Theorem 2.4. Thus we conclude that T 6= id−iH + i and T 6= id−iH − i. For the remaining cases we have that S = T. From Lemma 2.3 we conclude that there exists a biholomorphic map τ : Ω → Ω such that T(τ(z)) = τ(T(z)) and Ψ(z) = τ(Φ(z)). For the case when T = idC + 1 we obtain that there exists b ∈ C such that τ(z) = z + b for every z ∈ C. Since, for each z ∈ G ∩ R, 2iImΦ(x) = Φ(x) − Φ(x) = Φ(x) − Ψ(x) = Φ(x) − τ(Φ(x)) = −b ∈ iR, we see that, by passing over to Φ + b/2 instead of Φ, we may assume that Φ(x) ∈ R for each x ∈ G ∩ R. For the case when p ∈ F(g) and T = g0 (p)idC we obtain that there exists a ∈ C{0} such that τ(z) = az for each z ∈ C. Since, for each z ∈ G ∩ R, Φ(x) = Ψ(x) = aΦ(x), we see that, by passing to √ aΦ instead of Φ, we may assume that Φ(x) ∈ R for each x ∈ G ∩ R. For the case when there exists λ 1 such that T = λidiH we obtain that there exists c 0 such that τ(z) = cz for each z ∈ −iH. Since |Φ(x)| = |Φ(x)| = |Ψ(x)| = c|Φ(x)| for each x ∈ G ∩ R we conclude that c = 1. Hence we have that Φ(x) ∈ R for each x ∈ G ∩ R. Thus in any case we have that Φ(x) ∈ R for each x ∈ G ∩ R. We can now show that (∗) there exists σ ∈ {1, −1} such that Φ(H) ⊂ σH and Φ(−H) ⊂ −σH. Proof of (∗). First, we prove that there exists an unbounded component J of I ∩ V such that g(J) ⊂ J. This is clear for the case when p ∈ I. Hence we may assume that I ⊂ R. Let x ∈ I. Let K be the compact interval in R with endpoints x and g(x). Since K ⊂ I there exists n ∈ N such that gn (K) ⊂ I ∩ V . Since g(I ∩ V ) ⊂ I ∩ V it is easy to see that the component J of gn (K) in I ∩ V is unbounded such that g(J) ⊂ J. Now let x ∈ G such that Φ(x) ∈ R. We have that Φ(J) is an open interval in R such that T(Φ(J)) ⊂ Φ(J) and 0 ∈ Φ(J) if T = g0 (p)idC. In any case this implies that there exists n ∈ N such that Φ(gn (x)) = Tn (Φ(x)) ∈ Φ(J)
- 40. 16 DETLEF BARGMANN and gn (x) ∈ V . Let y ∈ J such that Φ(gn (x)) = Φ(y). Since g is injective on V we conclude that gn (x) = y which implies that x ∈ R because g is an inner function. Hence we have that Φ−1 (R) ⊂ R. This implies (∗). Next, we prove that there exists a fundamental set D for g on H such that Φ(D) is a fundamental set for T on Ω ∩ σH. To this end let z ∈ H and let K be a compact connected subset of H such that {z, g(z)} ⊂ K. Then there exists n ∈ N such that gn (K) ⊂ V ∩ H. Let D be the component of V ∩ H which contains gn (K). Since g(V ∩ H) ⊂ V ∩ H we conclude that g(D) ⊂ D. Since V is simply connected we have that D is simply connected. Let y ∈ H. Then there exists a compact connected set L ⊂ H such that {z, y} ⊂ L. Choose m ≥ n such that gm (L) ⊂ V ∩ H. Since gm (z) ∈ D we conclude that gm (L) ⊂ D and hence gm (x) ∈ D. Thus D is a fundamental set for g on H. Let u ∈ Ω∩σH. Since Φ(V ) is a fundamental set for T on Ω we find k ∈ N such that Tk (u) ∈ Φ(V ) ∩ σH. According to (∗) we find v ∈ V ∩ H such that Φ(v) = Tk (u). Choose l ∈ N such that gl (v) ∈ D. Then we have that Tk+l (u) = Tl (Φ(v)) = Φ(gl (v)) ∈ Φ(D). Hence Φ(D) is a fundamental set for T on Ω ∩ σH. Let E := Ω∩σH. We obtain that (E, T|E, Φ|H, D) is an eventual conjugacy for g on H. For the case when g|G ∼ idC + 1 we conclude that g|H ∼ idσH + 1. For the case when p ∈ I and g|G ∼ g0 (p)idC we conclude that g|H ∼ g0 (p)idH. Finally, suppose that there exists λ 1 such that g|G ∼ λid−iH. Let φ(z) := Φ(z)2 for each z ∈ H. Then we have that φ(g(z)) = Φ(g(z))2 = λ2 Φ(z)2 = λ2 φ(z) for each z ∈ H. Moreover, it is easy to see that φ is injective on D and φ(D) is a fundamental set for λ2 idσH on σH. Thus (σH, λ2 idσH, φ, D) is an eventual conjugacy for g on H. Hence g|H ∼ λ2 idH. 2 Moreover, we make use of the following two lemmata. Lemma 2.27. Let g be a non-Möbius inner function such that J (g) 6= ∂D. Suppose that there exists an invariant component I of F(g) ∩ ∂D, and let G := D ∪ I ∪ (Ĉ D). Let Ω ⊂ C be a simply connected domain and T be a biholomorphic self-map of Ω such that g|G ∼ T. If I is an absorbing component of F(g) ∩ ∂D, then g|F(g) ∼ T. Proof. Let V ⊂ G be a simply connected domain and let φ : G → Ω be a holomorphic function such that (Ω, T, Φ, V ) is an eventual congugacy for g on G. Since I is an absorbing component of F(g) ∩ ∂D it is clear that V is a fundamental set for g on F(g). It is easy to see that Φ(z) := T −n (φ(gn (z))) if gn (z) ∈ V is a well defined holomorphic function on F(g) such that Φ|G = φ. By the identity theorem we have that Φ ◦ g|F(g) = T ◦ Φ. Hence (Ω, T, Φ, V ) is an eventual conjugacy for g on F(g). 2
- 41. ITERATION OF INNER FUNCTIONS 17 The next lemma is proved in [1, p. 116, Lemma 2]. Lemma 2.28. Let γ be a closed rectifiable curve in C such that γ−1 (H) 6= ∅ 6= γ−1 (−H) and γ−1 (R) consists of exactly two points a and b. Then the open interval with endpoints γ(a) and γ(b) is contained in int(γ). Now, we are able to prove Theorem 2.22 and Theorem 2.23. Proof of Theorem 2.22. (2) ⇒ (1) and ((4) or (5) or (6)). Suppose that there exists an absorb- ing component I of F(g) ∩ ∂D. By Lemma 2.14, the set I is invariant under g. Let G := D ∪ I ∪ (Ĉ D). By Lemma 2.26, exactly one of the following statements holds: (40 ) p ∈ F(g) and g|G ∼ g0 (p)idC and g|D ∼ g0 (p)idH, (50 ) p ∈ J (g) and g|G ∼ idC + 1 and g|D ∼ idH ± 1, (60 ) p ∈ J (g) and there exists λ 1 such that g|G ∼ λidH and g|D ∼ λ2 idH. Since I is absorbing we conclude from Lemma 2.27 that (40 ) implies (4), (50 ) implies (5), and (60 ) implies (6). (3) ⇒ (2). Suppose that there are no wandering components of F(g)∩∂D. Since F(g) ∩ ∂D 6= ∅ we conclude from Lemma 2.14 that there exists an invariant component I of F(g)∩∂D. Let J be a component of F(g)∩∂D. By Lemma 2.14 there exists n ∈ N such that gn (J) is contained in an invariant component of F(g) ∩ ∂D. By Lemma 2.13 there is at most one invariant component which implies that gn (J) ⊂ I. Hence I is an absorbing component of F(g) ∩ ∂D. (1) ⇒ (3). Suppose that there exists an eventual conjugacy (Ω, T, Φ, V ) of g on F(g). We may assume that p ∈ J (g) because otherwise the component I of p in F(g) ∩ ∂D is absorbing and the conclusion follows. By conjugating with an appropriate transformation we may further assume that g is an inner function of H and p = ∞. Let I be a component of F(g) ∩ R. Let x ∈ I and let γ be the positively oriented circle centered at (x + g(x))/2 which contains x and g(x). Then γ is a compact subset of F(g) and there exists n ∈ N such that gn (γ) ⊂ V . Let J be the component of F(g) ∩ R which contains gn (x). Since V is simply connected and gn ◦ γ is a closed analytic curve in V we conclude that int(gn ◦γ) ⊂ V . From Lemma 2.28 we see that the open interval K with endpoints gn (x) and gn+1 (x) is contained in int(gn ◦ γ) ⊂ V ⊂ F(g). Hence K ⊂ J and {gn (x), gn+1 (x)} ⊂ J. This implies that I is not wandering. 2 Proof of Theorem 2.23. (1) ⇒ (2). Suppose that g|F(g) ∼ idC + 1. Then g does not have a fixed point in F(g) which implies that p ∈ J (g). From Lemma 2.6 we know that λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞ for every z ∈ F(g). (2) ⇒ (3). Suppose that p ∈ J (g) and λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞ for some z ∈ F(g). By conjugating appropriately we may assume
- 42. 18 DETLEF BARGMANN that g is an inner function of H and p = ∞. Since J (g) is perfect, F(g) is a hyperbolic domain in C such that F(g) has no isolated boundary points. Let x ∈ F(g) ∩ R and define φn : F(g) → C, z 7→ gn (z) − gn (x) gn+1(x) − gn(x) = Tn(gn (z)), where Tn(w) := (w − gn (x))/(gn+1 (x) − gn (x)). From Theorem 2.10 we know that the sequence (φn)n∈N converges locally uniformly on F(g) to a holomor- phic function φ : F(g) → C such that φ(g(z)) = φ(z) + 1 for all z ∈ F(g). Let G := φ(F(g)). We first prove that (∗) [ n∈N (G − n) = C. Proof of (∗). We may assume that G is hyperbolic because otherwise CG contains at most one point and (∗) follows immediately. Since φ◦g|F(g) = φ+1 we have that (idC + 1)(G) ⊂ G. Moreover, from the Schwarz-Pick lemma we conclude that λG(φ(z) + n, φ(z) + n + 1) = λG(φ(gn (z), φ(gn+1 (z))) ≤ λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞ for some z ∈ F(g). Thus we can apply Theorem 2.11 and obtain that (∗) holds. Since G+1 ⊂ G, the sequence (G−n)n∈N is increasing, which implies that there exists k ∈ N such that [0, 1] ⊂ G−k, and hence [φ(gk (x)), φ(gk+1 (x))] = [k, k + 1] ⊂ G ∩ R = φ(F(g)) ∩ R. Since φn → φ locally uniformly in F(g) we find n0 ∈ N such that, for each n ≥ n0, [φn(gk (x)), φn(gk+1 (x))] ⊂ φn(F(g)) ∩ R = φn(F(g) ∩ R). Hence, for each n ≥ n0, [Tn(gn+k (x), Tn (gn+k+1 (x))] ⊂ Tn(F(g) ∩ R), which implies that the closed interval with endpoints gk+n (x) and gk+n+1 (x) is contained in F(g) ∩ R for every n ≥ n0. Let I be the component of F(g) ∩ R which contains u := gk+n0 (x). Then g(u) ∈ I and hence g(I) ⊂ I which implies that I is unbounded because (g|F(g))n → ∞ as n → ∞. Moreover, g|I does not have any fixed point which implies that either g(z) z for each z ∈ I and sup I = ∞ or g(z) z for each z ∈ I and inf I = −∞. We show that I is an absorbing component of F(g) ∩ R. To this end let y ∈ F(g)∩R. Then φn(y) ∈ R, for each n ∈ N, and hence φ(y) ∈ R. Thus we find m ∈ N such that φ(gm (y)) = φ(y)+m 1. Hence there exists n ≥ n0 +k such that gn (gm (y)) − gn (x) gn+1(x) − gn(x) = φn(gm (y)) ≥ 1.
- 43. ITERATION OF INNER FUNCTIONS 19 Let l := n − n0 − k. If now g(z) z, for each z ∈ I, then gn+1 (x) = gl+1 (u) gl (u) = gn (x), which implies that gn (gm (y)) gn (x) = gl (u) and hence gn+m (y) ∈ I because sup I = ∞. We can argue analogously for the case when g(z) z for each z ∈ I. Thus gn+m (y) ∈ I and I is absorbing. It remains to show that gH ∼ id±H + 1. To this end, first note that φn(H) ⊂ H or φn(H) ⊂ −H for each n ∈ N. Since φ is not constant we conclude that there exists σ ∈ {−1, 1} such that φ(H) ⊂ σH. According to Theorem 2.4 there exists a fundamental set V ⊂ H for g on H. Since g(V ) ⊂ V and g|V is injective we have that φn|V is injective, for each n ∈ N, and hence φ is injective on V . Since φ(g(z)) = φ(z) + 1 for every z ∈ H it remains to show that W := φ(V ) is a fundamental set for idσH + 1. Let w ∈ σH. According to (∗) there exists q ∈ N such that w+q ∈ G∩σH = φ(H). Choose z ∈ H such that w + q = φ(z). Since V is a fundamental set for g on H there exists r ∈ N such that gr (z) ∈ V and hence w + q + r = φ(z) + r = φ(gr (z)) ∈ φ(V ) = W. Thus W is a fundamental set for idσH + 1 and g|H ∼ idσH + 1. This proves (3). The implication (3) ⇒ (4) follows immediately from Lemma 2.14. To prove (4) ⇒ (1) suppose that g|D ∼ idH ± 1 and there exists an invariant component I of F(g) ∩ ∂D. Let G := D ∪ I ∪ (Ĉ D). From Lemma 2.26 we conclude that p ∈ J (g) and g|G ∼ idC + 1. By Lemma 2.6 we have that λG(gn (z), gn+1 (z)) → 0 as n → ∞, for each z ∈ G, which by the Schwarz- Pick lemma implies that λF(g)(gn (z), gn+1 (z)) → 0 as n → ∞ for every z ∈ G. Thus (2) holds and we conclude from the implication (2) ⇒ (3) that there exists an absorbing component of F(g)∩∂D. Since each absorbing component is invariant and there is at most one invariant component of F(g) ∩ ∂D we have that I is absorbing. Thus we can apply Lemma 2.27 and obtain that g|F(g) ∼ idC + 1. 2 2.4. Periodic points of an inner function. It is well-known that the Julia set of a non-injective rational, entire, or meromorphic function f is equal to the closure of the set of repelling periodic points of f. In this subsection we see that the same is true for inner functions if the definition of a repelling periodic point is relaxed in a certain sense. We make use of the following theorem which is known as the Julia-Wolff lemma [18, Proposition 4.13]. For the definition and the main properties of the angular derivative we refer the reader to [18].
- 44. 20 DETLEF BARGMANN Theorem 2.29 (Julia-Wolff). Let h be a holomorphic self-map of the unit disk. Let ζ, η ∈ ∂D such that h has the angular limit η at ζ. Then the angular derivative h0 (ζ) exists in (0, ∞] and satisfies h0 (ζ) = η ζ sup z∈D 1 − |z|2 |ζ − z|2 |η − h(z)|2 1 − |h(z)|2 . Definition 2.30. Let h be a holomorphic self-map of the unit disk. Let ζ ∈ ∂D. (1) We call ζ a fixed point of h if h has the angular limit ζ at ζ. In this case the fixed point ζ of h is called repelling if the angular derivative h0 (ζ) is contained in (1, ∞]. (2) We call ζ a (repelling) periodic point of h if there exists n ∈ N such that ζ is a (repelling) fixed point of hn . Theorem 2.1 represents only one part of the theorem of Denjoy and Wolff. The second part of the Denjoy-Wolff theorem (see [15] and [22]) is a statement about the angular derivatives at the fixed points of a holomorphic self-map of D: Theorem 2.31 (Denjoy-Wolff). Let h be a non-Möbius holomorphic self-map of the unit disk. Let p ∈ D be the Denjoy-Wolff point of h. Then p is the unique fixed point of h which is not repelling. Hence there is at most one periodic point of a non-Möbius inner function g which is not repelling. Since J (g) is perfect it remains to show that the periodic points of g are dense in the Julia set. This is a consequence of the following theorem. Theorem 2.32. Let g be an inner function. Suppose that ζ ∈ ∂D is a singularity of g. Then ζ is an accumulation point of fixed points of g. We use the following lemma to prove Theorem 2.32. Lemma 2.33. Suppose that g is a non-Möbius inner function of the upper half plane H. Then the following statements are equivalent. (1) ∞ is the Denjoy-Wolff point of g. (2) Im(g(z)) ≥ Im(z), for each z ∈ H. (3) G(z) := g(z) − z is an inner function of the upper half plane. (4) Im(g(z)) Im(z), for each z ∈ H. Proof. Suppose that (1) holds. Let T(z) := i(1 + z)/(1 − z). Then h := T−1 ◦ g ◦ T is an inner function of the unit disk with Denjoy-Wolff point 1. An easy calculation shows that Im(T(z)) = (1 − |z|)2 |1 − z|2
- 45. ITERATION OF INNER FUNCTIONS 21 for each z ∈ D. Since the angular derivative h0 (1) is contained in (0, 1] we conclude from Theorem 2.29 that Im(g(z)) ≥ Im(z), for each z ∈ H. Suppose that (2) holds. Let G(z) := g(z) − z. Then Im(G(z)) ≥ 0, for each z ∈ H, which implies that G is an inner function of H or G is constant. Since g is non-Möbius we conclude that G is non-constant and hence an inner function. The implication (3) ⇒ (4) is trivial. Suppose that (4) holds. It is easy to see that, for each p ∈ H ∪ R, there exists z ∈ H such that (gn (z))n∈N does not converge to p. Hence we conclude from the Denjoy-Wolff theorem that ∞ is the Denjoy-Wolff point of g. 2 Proof of Theorem 2.32. Let p ∈ D be the Denjoy-Wolff point of g. Case 1. Suppose that p ∈ D. By conjugating with a Möbius transformation if necessary we may assume that p = 0. It is easy to see that h(z) := g(z) z is an inner function with a singularity at ζ. Let U be a neighbourhood of ζ. By Lemma 1.5 there exists θ ∈ (U ∩ ∂D) {ζ} and a path γ : [0, 1) → D such that limr→1 γ(r) = θ and h(γ(r)) → 1 as r → 1 in a Stolz angle. Hence we have that g(γ(r)) → θ as r → 1. By the Lehto-Virtanen theorem [18, p. 71], we conclude that g has angular limit θ at θ. Case 2. Suppose that p ∈ ∂D. We may assume that p = 1. Let T(z) := i(1 + z)/(1 − z). Then G := T ◦ g ◦ T−1 is an inner function of the upper half plane with Denjoy-Wolff point ∞ and a singularity at T(ζ). From Lemma 2.33 we conclude that H(z) := G(z) − z is an inner function of the upper half plane which has a singularity at T(ζ). Let U be a neighbourhood of T(ζ). By Lemma 1.5 there exists θ ∈ (U ∩ R) {ζ} and a path γ : [0, 1) → H such that limr→1 γ(r) = θ and H(γ(r)) → 0 as r → 1 in a Stolz angle. Hence we have that G(γ(r)) → θ as r → 1. By the Lehto-Virtanen theorem again, we conclude that G has angular limit θ at θ. 2 Theorem 2.34. Let g be a non-Möbius inner function. Then J (g) is the closure of the set of the repelling periodic points of g. Proof. For the case when g is a finite Blaschke product this is an old theorem of G. Julia and P. Fatou; see for instance [6, Theorem 6.9.1] or [20], [5] and [10] for simpler proofs. For the case when g has a singularity on ∂D it follows from Theorem 1.4 and Theorem 2.32 that J (g) is contained in the closure of the set of the periodic points of g. Since there is at most one periodic point of g which is not repelling and J (g) is perfect we conclude that even the repelling periodic points of g are dense in J (g). On the other hand, the only periodic point of g which might be in F(g) is the Denjoy-Wolff point. Hence all repelling periodic points are contained in J (g). 2.
- 46. 22 DETLEF BARGMANN 2.5. Examples. In this subsection we give examples to see that all cases listed in Theorem 2.22 may occur (see Examples 2.36, 2.37, and 2.39). More- over, we give an example of an inner function g of the upper half plane having an invariant component of F(g) ∩ (R ∪ {∞}) which is not absorbing (see Ex- ample 2.38). Most of the examples of inner functions in this subsection are of the fol- lowing type. Let Q be an at most countable subset of R. Let w : Q → (0, ∞) be such that P q∈Q w(q) converges. Let a ∈ [0, ∞) and let b ∈ R. Then it is easy to see that g(z) := az + b − X q∈Q w(q) z − q is an inner function of the upper half plane. If Q is an infinite set, then the singularities of g are the accumulation points of Q. If dist(z, Q) is large, then g(z) is nearly az + b. In particular, this is the case when |Im(z)| is large. Moreover, it follows from Lemma 2.33 that ∞ is the Denjoy-Wolff point of g if and only if a ≥ 1. First, we discuss some examples of inner functions g on the upper half plane which have the property that g|H ∼ λidH, for some λ 1. We make use of the following lemma. Lemma 2.35. Let g be a non-Möbius inner function of the upper half plane. Suppose that there exists r 1 such that Im(g(z)) rIm(z), for each z ∈ H. Then ∞ is the Denjoy-Wolff point of g and there exists λ 1 such that g|H ∼ λidH. Proof. Since Im(g(z)) ≥ Im(z), for each z ∈ H, we conclude from Lemma 2.33 that the Denjoy-Wolff point of g must be ∞. Moreover, for each z, w ∈ H, we have that λH(z, w) = log 1 + |(z − w)/(z − w)| 1 − |(z − w)/(z − w| , which in particular implies that λH(z, w) ≥ λH(iIm(z), iIm(w)). Now, for each z ∈ H, |Im(g(z)) − Im(z)| |Im(g(z)) − Im(z)| = Im(g(z)) − Im(z) Im(g(z)) + Im(z) = 1 − 2Im(z) Im(g(z)) + Im(z) ≥ 1 − 2Im(z) (r + 1)Im(z) = 1 − 2 r + 1 , which implies that inf{λH(z, g(z)) : z ∈ H} 0.
- 47. ITERATION OF INNER FUNCTIONS 23 From Theorem 2.4 and Lemma 2.6 we conclude that there exists λ 1 such that g|H ∼ λidH. 2 Example 2.36. Let Q be a countable subset of [0, ∞) which is dense in [0, ∞). Let w : Q → (0, ∞) be such that P q∈Q w(q) converges. Let a 1 and b ∈ R. Then g(z) := az + b − X q∈Q w(q) z − q is an inner function of the upper half plane with the following properties. (1) There exists an absorbing component of F(g) ∩ (R ∪ {∞}). (2) The Denjoy-Wolff point ∞ of g is contained in J (g). (3) There exists λ 1 such that g|F(g) ∼ λidH and g|H ∼ λ2 idH. Proof. Since Q is bounded from the left we conclude that there exists c 0 such that Re(g(z)) −c whenever Re(z) −c. Hence (−c, ∞) is contained in an invariant component I of the Fatou set of g. Since Q is dense in [0, ∞) we conclude that [0, ∞) ∪ {∞} ⊂ J (g). Let J be another component of F(g) ∩ R. Then fn (J) ⊂ F(g) ∩ R ⊂ (−∞, 0), for each n ∈ N, and fn → ∞ locally uniformly on J as n → ∞. This implies that there exists n ∈ N such that fn (J) ⊂ I. Thus I is absorbing. From Lemma 2.35 we conclude that there exists µ 1 such that g|H ∼ µidH. Now, the conclusion follows from Theorem 2.22 . 2 Example 2.37. Let Q be a bounded subset of R which is not empty and at most countable. Let w : Q → (0, ∞) be such that P q∈Q w(q) converges. Let a 1 and b ∈ R. Then g(z) := az + b − X q∈Q w(q) z − q is an inner function of the upper half plane with the following properties. (1) There exists an absorbing component of F(g) ∩ (R ∪ {∞}). (2) The Denjoy-Wolff point ∞ of g is contained in F(g). (3) g|F(g) ∼ 1 a idC and g|H ∼ 1 a idH. Proof. Since Q is bounded there exists c 0 such that Re(g(z)) −c whenever Re(z) −c, and Re(g(z)) c whenever Re(z) c. This implies that there exists an invariant component I of F(g)∩(R∪{∞}) which contains the Denjoy-Wolff point ∞. Hence I is absorbing and the conclusion follows from Theorem 2.22 and the fact that the eigenvalue of the fixed point ∞ of g is equal to 1/a. 2
- 48. 24 DETLEF BARGMANN Example 2.38. The function g(z) := 2z − X n∈N 2−n z − 2n is an inner function of the upper half plane with the following property: g has a wandering and an invariant component of F(g)∩(R∪{∞}). In particular, the invariant component is not absorbing. Proof. Since Q := {2n : n ∈ N} is bounded from the left there exists c ∈ (0, ∞) such that Re(g(z)) −c whenever Re(z) −c. This implies that there exists an invariant component I of F(g) ∩ (R ∪ {∞}). Since ∞ is a accumulation point of Q we conclude that ∞ ∈ J (g). Choose 1 a b 2 such that x(x − 1) ≥ 1 2 and x(2 − x) ≥ 1 2 , for each x ∈ [a, b]. We prove: (∗) ∀ a ≤ α β ≤ b, ∀ m ∈ N, ∀ x ∈ [2m α, 2m β] : 2m+1 α(1 − 2−m ) ≤ g(x) ≤ 2m+1 β(1 + 2−m ). Proof of (∗). Let a ≤ α β ≤ b, m ∈ N, and x ∈ [2m α, 2m β]. Then − X n∈N 2−n x − 2n ≤ X n≥m+1 2−n 2n − x ≤ X n≥m+1 2−n 2n − 2mβ = 4−m X n∈N 2−n 2n − β ≤ 4−m (2 − β)−1 . This implies that g(x) ≤ 2x + 4−m (2 − β)−1 ≤ 2m+1 β + β4−m (β(2 − β))−1 ≤ 2m+1 β(1 + 2−m ). On the other hand, − X n∈N 2−n x − 2n ≥ − X n≤m 2−n x − 2n ≥ − X n≤m 2−n 2mα − 2n ≥ − X n≤m 2−n 2mα − 2m ≥ −2−m (α − 1)−1 , which implies that g(x) ≥ 2x − 2−m (α − 1)−1 ≥ 2m+1 α − α2−m (α(α − 1))−1 ≥ 2m+1 α(1 − 2−m ). Thus we have proved (∗). Since Q m∈N(1 − 2−m ) and Q m∈N(1 + 2−m ) are convergent, we can find a α0 β0 b and m0 ∈ N such that α0 Y m≥m0 (1 − 2−m ) ≥ a and β0 Y m≥m0 (1 + 2−m ) ≤ b.
- 49. Other documents randomly have different content
- 53. The Project Gutenberg eBook of Scrambled World
- 54. This ebook is for the use of anyone anywhere in the United States and most other parts of the world at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this ebook or online at www.gutenberg.org. If you are not located in the United States, you will have to check the laws of the country where you are located before using this eBook. Title: Scrambled World Author: Basil Wells Release date: December 29, 2020 [eBook #64172] Most recently updated: October 18, 2024 Language: English Credits: Greg Weeks, Mary Meehan and the Online Distributed Proofreading Team at http://www.pgdp.net *** START OF THE PROJECT GUTENBERG EBOOK SCRAMBLED WORLD ***
- 56. Redskins! Boston tealeggers! Jeep men! Time traveler Devin Orth clutched his temples, battling insanity. Some genius had waved a wand over Terran history and produced a— Scrambled World
- 57. By BASIL WELLS [Transcriber's Note: This etext was produced from Planet Stories Spring 1947. Extensive research did not uncover any evidence that the U.S. copyright on this publication was renewed.]
- 58. The sun was dying. About its sullen shadow-streaked red globe thousands of miniature artificial worlds clustered like a swarm of night-chilled midges. So thickly did they hug the great globe of dulling flame that it seemed Sol had acquired an outer husk of interlocked asteroids and moonlets. Of all the planets and their satellites only Earth remained—a shrunken and changed planet. And Earth too had shifted its orbit until it now swung but a few million miles from its molten primary. In the huge ovoid of metal that was the Time Bubble the three men making up its crew had by now grown accustomed to the changes that three million years had brought to the solar system. They had expected great changes—and found them. This was to be their first stop in their time quest for an efficient shield against the deadly radiations of atomic disintegration's side effects. Devin Orth, the lean, dark-haired young scientist sharing the control blister with his employer and friend, Norris Horn, studied the expanding green wilderness of what had once been northern Ohio. He turned to the big bald man in whose brain the plans for the time spanner had been born. The continents are there, he said unbelieving, almost exactly as we left them. And yet Earth is smaller. Its diameter has dwindled more than a thousand miles! Horn's broad thick body quivered as he loosed a volcanic chuckle. I know, he said. And the oceans, big though they are, are probably very shallow. A thousand feet at the most. Water will be growing precious. But, puzzled Orth, why are there no cities and why have the continents changed so little? Surely three million years.... I'd say the inhabitants of those small globes near the sun, suggested Horn, are descendants of Earthmen. They have used
- 59. their superb command of science to make of Earth a beautiful park or preserve as it was in our own primitive age. Surely, if they have such knowledge, they can give us the secret of atomic control that will overcome the sterility threatening mankind. We cannot return now to the limited culture afforded by the lesser power-sources of coal or gas without great damage to civilization— perhaps its utter downfall. They have it all right, said Orth, scowling down at the open parklike meadow toward which Horn was blasting, but I'm worried about getting back. So far this time travel is simply negation— outside the Time Bubble three million years pass and to us it seems less than two hours. Horn thrummed the landing jets smoothly and laughed his deep booming bellow. The grassy glade came up to meet them. A minor detail, he said as he cut the jets and the ship jolted abruptly to an uneven grounding. The deck was slightly tilted and from below there sounded a muffled explosion. You all right, Neilson? shouted Orth into the intercom. The third member of their crew sounded breathless as he answered from the power compartment. Thought the mixer was going for a bit, he gasped. A forward jet went kafoo. Boulder maybe blocking off that last blast. Orth told Horn what Neilson had said. The big man unzipped his safety harness and came over to his side, his big capable hand on Orth's shoulder. Don't worry about getting home, he said, taking up the thread of conversation the explosion had disrupted. In three million years all the secrets of time and matter will have been discovered. We'll return with the shield. He released the young scientist's bruised shoulder and slapped a great paw of a hand on his back, pushing him down toward the
- 60. airlock. Better replace that jet tip, Devin, he said. Can't tell but we may have to take off in a hurry. This future civilization might be unfriendly and, he paused thoughtfully, even non-humanoid. Orth checked the gauges at the lock and found the outer atmosphere to be a heady oxygen-rich mixture. Horn had gone down to help Neilson in the power compartment and he was alone. He stuffed the jet tip into his bag of tools and pushed through the inner port into the airlock. There he snapped on the invisible, but oddly tingling, radiations that would destroy any alien spores of deadly growth that might find their way into the ship. He swung open the thick oval outer door and dropped the short grounding ladder to the blast-blackened turf. Down the eight rigid metal rungs of the ladder he went to the ground. He stumbled awkwardly and almost fell. The unaccustomed gravity, after the past twelve days in space—twelve days that had actually been thirty thousand centuries—had tricked him. A moment later his muscles had quickly remedied this unbalance and he found the fused jet that had blown back. As Neilson had guessed, the Time Bubble had grazed a boulder in landing and the expanding rocket gases' escape had been blocked off. It was good to feel the spring of turf underfoot. Even the feeble warmth of the ancient sun was pleasant on his bared flesh. He had not realized how homesick he had grown for Earth until now. He put down his tools and headed toward a clump of oddly-shaped trees near the forest's rim. As he neared them he whistled. The temperature of the Lakes region must have changed. They were palms! It was only then that he turned to look back at the Time Bubble. He was thinking that Horn would be interested in his discovery of this tropical growth so far north. His eyes blinked stupidly. He blinked again.
- 61. The Time Bubble's ugly ovoid of space-scarred metal was gone! Several hours had passed since the space ship's uncanny disappearance. The Earthman was picking his way along a narrow game trail in the semi-twilight of the mighty forest that crowded close up to Lake Erie's shoreline. Caution had impelled him to seek safety in the wilderness until the truth about the spacer's disappearance was revealed. The trail cut across a rock-strewn highway, deeply-rutted by wheeled vehicles. Just across the way, half-hidden by a tangle of wild vines and brush, was a small log cabin. Smoke oozed slowly skyward from its mud-daubed stick chimney. The odor of cooking meat sent Orth trotting hungrily across the road. He had forgotten any possible danger until an arrow hissed viciously past his ear. He dropped forward on his belly in a shallow depression soggy with dead leaves. A second arrow thwocked lightly through the gray-barked tangle of brush that his head was ramming into. His fingers went to the flat pocket machine gun that all three scientists aboard the Time Bubble carried. This weapon, complete with ten thousand tiny explosive cartridges, and a compact kit of tools and essential equipment, they carried with them at all times when away from the space ship. Behind a light gray shaft of scaly bark, a huge tree's bole, something red moved. His machine gun slapped a dozen needle-sized slugs at the half-seen target. The explosions splintered and ripped at the tree's thick trunk. The red thing leaped clear, yelling. Before Devin could stop his weapon, it stepped into several small incredibly bright explosions.
- 62. Before Devin could stop firing, the thing leaped clear, yelling. Then, from the cabin, a broad-shouldered young man emerged. He was clad somewhat after the fashion of the early American pioneers: fur cap, shapeless brown homespun shirt, rough skinny-legged trousers, and thick-soled moccasins. In his hands he lifted a cumbersome weapon, having six wooden barrels or tubes, from each of which protruded a sharp-pointed metal dart. There been trouble? he cried out in badly mangled but understandable English.
- 63. For an instant Orth was stunned by the wonder of it. After three million years—a man speaking English! Shot at me from over there, he told the frontiersman warily. The man catfooted over to the scarred tree, his clumsy weapon poised ready. He grunted something in badly garbled English. Then he motioned to the Earthman to join him. Redskin, he told Orth. The hairy apelike savage crouching in bloody death behind the tree was indeed clad in flapping, red-dyed garments of skin. His skin, however, was as white beneath its matted covering of black hair as Orth's own. Yet the other had called the savage a redskin. As Orth watched the tall young giant stamped his foot down on the fallen warrior's middle, shook the long chestnut hair out of his handsome brown face, and opening his mouth let out a prolonged hideous screech. As he did so his fists hammered drumlike on his distended chest. From the distance a hideous snarling and trumpeting answered the ear-splitting sound. The man grinned at Orth and nodded toward the forest. He stepped down and held up two fingers. Vello, he said, continuing to make the V sign that first saw birth in the Second World War. Me, I am Dun Horgan. Horgan of the wilderness. Those are my friends you hear, the hairy apes of Afri County. Orth held out his hand. Shake, he said, Horgan. I'm named Orth. I hail from Meadville in Pennsylvania. Pennsylvania over that way, and Horgan pointed, but no village that name. Maybe small? Orth nodded. Small, he agreed wryly. After three million years he wondered that the states retained their original names. Horgan reached down to jerk an intricately woven necklace of hair, from which depended a crudely carved locket of bone, from the
- 64. fallen savage's neck. Scalp locket is worth fifteen bits bounty, he said offering it to Orth. It is yours. Orth shook his head. No, you keep it. I'll trade it for some food and a bed. He eyed the other thoughtfully. And some information too, he added. Over a well cooked slab of venison and a plate of corn bread, washed down by a muddy brown brew that Horgan served hot and sweetened, they talked. Corn likker the frontiersman called the steaming tasteless fluid when Orth mistakenly named it coffee. And when they had finished his host produced squares of a fine brown paper which he deftly filled, one-handed, with shredded greenish tobacco, and presented the fat cigar-sized bundles to Orth. He shrugged at the Earthman's refusal, eyeing with amusement the slender whiteness of Orth's own cigarettes. Shipped from France maybe, he suggested, or China? Orth handed over the pack. Horgan studied the markings that showed they were manufactured in Kentucky. He shook his head. Don't reckon you'll be getting no more, he said. General Lee ain't been licked yet, and until Washington and Pershing break through to the South.... He lifted his big arms in a half-shrug of doubt. What's all this about Lee and Pershing? Some sort of Civil War over again? Or is this continent being invaded? Horgan eyed the Earthman curiously. Maybe I'll have to tell you what year it is, he said dryly, and who's Boss of the States now. You're powerful ignorant, Orth. Go ahead, invited Orth. My memory's fuzzy.
- 65. This's 1927, June the third. Horgan tugged absent-mindedly at his long brown locks. Our boss now is Tyad Roosfald. His third year as Boss. Teddy Roosevelt. Orth studied his knuckles thoughtfully. And I suppose General Eisenhower is invading Germany to win our independence! Not Germany, corrected the frontiersman, but Great Britain. We have accepted Churchill's challenge to land there and fight. Of course the war with Germany and Japan are going on too. Orth groaned. What about this other thing—Lee and Washington down South? Don't tell me it's Civil War revival week too. What kind of a gag are you trying to pull on me, Horgan? Horgan rubbed a rasping palm thoughtfully along his jaw. There is war between the States, he said at last. Everywhere there is war. The broadcast drums warn us that soon we must fight Cuba. Smoke puffed from his nostrils. Helping Spain. One of Orth's hands covered his eyes and he felt his face growing hot with a mingling of anger and bewilderment. He stuttered as he tried to talk. He swallowed smoke and coughed, choking. Good afternoon, called a fresh young voice, a feminine voice, from the cabin's rude door. Their heads twisted smartly toward the opening. Horgan's bared sword was in his fist even before he was on his feet. Together they stood facing the tall round-bodied woman who had walked in upon them. Despite her stature she was remarkably beautiful. She was pale of skin and her great mass of intricately braided hair was of a softly radiant silver hue. The simple garment of golden- hued cloth covered her adequately—but no more than that. Even her
- 66. sandals were simple, accessories of comfort and utility rather than fashion. I am Ayna of Globe 64BA, she told them briskly. I wish one or both of you to escort me to Ivath's headquarters. She was eyeing Orth's zippered shirt and glassid trousers curiously. Ivath must be slipping, she said. You are definitely out of the wrong century. More likely the Twenty-first. I cannot be mistaken for I have majored in Ancient American Mythology. I was born in 1960! Orth snapped, and I definitely must be in the wrong century. Or I'm out of my head! That's more like it. All this pother about the Civil War and the World Wars going on at the same time. Maybe just the names are the same. Or—what? There must be a short circuiting of your memory cells, said Ayna soothingly, but Ivath and his helpers will soon set that right. Take me to him and I will help you. She looked at Horgan. Horgan was shaking his head. Sorry, he said, but until the Civil War is ended—here I stay. The girl frowned. She turned to Orth. How about you? she demanded. Are you part of the local scenery too, or can you travel? I have no idea what this is all about, Orth told her, but I go where I please. Maybe you can set me right on a few things, Ayna. Then I'll go along with you. Fine! Her teeth flashed. I can go with you to Hardpan City, Dun Horgan said slowly. That's where I trade off my furs and gold dust. We can thump a ride on one of the waggons going to New Yok. What are we waiting for then? demanded Ayna. Bring extra slugs for your six guns. She looked at Orth. Don't you have a gun?
- 67. Orth tugged out his compact machine pistol. Apparently the clumsy spring-powered weapons with six barrels were what the girl called six guns, for Horgan belted a second weapon around his waist. The girl examined his hand gun with curious eyes and fingers. Unusual design, she commented. Not authentic for your period costume. Come along, said Horgan, cutting across Orth's protesting words. About time for the afternoon waggon train. Here they come! cried Horgan as they quitted the path for the rutted highway. He seized a long length of pole and started beating at the road with it. Dust clouded up about them. And further down the road a growing cloud of dust neared. These must be the waggons Horgan was going to hail, Orth decided. Why is he pounding the ground, Ayna? demanded Orth. The woman laughed. He is thumping for a ride, she explained. It is a peculiar custom of this age. In this way he asks assistance. Orth's dark face reddened with sudden mirth that he choked back. The twisted idiomatic expressions of this strange world were taking a familiar pattern. Even the scrambled pattern of wars and their military leaders began to make sense. Thumping a ride, six shooters, and scalp lockets linked up with Ayna's reference to Ancient American Mythology. You're from the little globes clustering around the sun, he said, and you were visiting Earth—or this primitive duplication of it. Sort of a park for your people, this. Your spacer crashed or you've lost it. Ayna frowned at Orth. Yes, she said slowly, I landed on Earth, contrary to the regulations, and a herd of mammoths wrecked my
- 68. ship. But how could you, a creature of Ivath's great workship, know anything of spacers? I do not know Ivath, Orth said angrily, and I came here in a spacer that has vanished.... Now, how do we get out of this make believe world of yours to your home? But this is real, the girl protested. If a redskin's arrow or a tearunner's slug cuts you down you will die. Until the war is ended, or you take me to Ivath's headquarters, we are not safe. All I can say is human beings are as crazy as they were three million years ago, grunted Orth. Meanwhile the dust cloud rolled closer and slowed. Horgan's thumping had halted them. Orth saw three great waggons, their twenty foot-high metal-tired wheels fitting deep down into the rutted way. Sixty feet in length they were, and beneath a low roof, that Ayna called a hood, there was a broad treadmill geared up with the eight huge wheels. Between eighty and a hundred thick-bodied little ponies were tied upon this raised moving belt. Above the hood lifted a sort of tower, its roof twenty feet above the ground, and here the two waggon drivers sat, steering the cumbersome vehicle with a spoked wooden wheel. Back of the cab was the covered cargo deck of the waggon where bags of grain, hides and other produce were heaped. One of the wooden blocks that had braked the enormous wheels was smoking and now it burst into flame. One of the drivers hastily tossed a bucket of water on the block and put it out. Going through Hardpan City? Horgan asked. Climb aboard, cheerfully answered a runty driver with a huge dusty red moustache. He jabbed his thumb at the ladder bolted to the waggon's side.
- 69. You ride this waggon, Horgan said to Ayna and Orth. I'll hop the next one. The red-moustached man helped them into the cab, his squinted pale eyes studying the girl appreciatively, and then he spoke to his hulking companion. This driver was a hairy apish giant without ears. Now he slowly released the brakes that locked the treadmill while Red Moustache freed the wheels. The treadmill revolved faster and faster and they went clanking and bumping off down the highroad, the miniature horses sweating in their involuntary struggle to keep on their feet. The great hooded vehicle had a pace of perhaps ten miles an hour. I hear, shouted the little driver at Orth and Ayna, that the Boss is sending a hundred men to New Yok soon. They're to hunt down the red jitterbugs and outlaws that range the highways. He paused long enough to catch his breath and curse the thick fog of dust that filled the cab. A hundred soldiers to wipe out three or four thousand tea sellers and their gunmen! He snorted. Of course they're jeep men— Hoovers, you know—but they can't do any good. The Boss is all wise, said the earless man, bumping his clenched fist against his nose. He is the Boss. Orth turned to Ayna. Now, he said, who is Ivath? The girl shrugged. For a creation of Ivath's laboratories, she said, you are refreshingly human. So I will treat you as one of us. Her eyes were thoughtful. After all a robot does possess a limited power of reasoning. Ivath! Orth barked the word at her. Forget the insults for the time being. I may look funny but I'm human. Ivath is the director of our theater of space, she said. This, as you know, is a huge hollow globe on whose surface world-wide dramas
- 70. from the ages past are brought to life. He is painstakingly accurate in his depiction of the bygone dress, customs and speech. Orth laughed shortly. Even to vehicles with horses for power, he said, and guns without gunpowder. The girl disregarded him. But Ivath has surpassed other directors of the past. He uses androids, living robots, and impresses on their memory cells the accurate thought and instinct patterns of their own chosen age. It is really amazing how closely their actions follow the historical patterns of the ancient past. You mean he sprinkles cities, forests and—robots, all around and watches what happens? No script for them to follow? No deadline or time to end it all? He usually changes the entire surface of the globe every fifty years, Ayna told him. The next drama will be that of ancient Mars before the Earthmen came, and shortly afterward. If it is as accurate as this mess, said Orth dryly, it will be something to see, and worse to hear. I lived in the years of the first Martian exploration, Ayna. And I came from the Twentieth Century that your director is supposed to be presenting here! Ayna's face was serious. Orth felt a curious prickling sensation in his head and then everything went hazy for a time.... Eventually the blur faded. He found that they had left the forest behind and were entering a region of cultivated fields and little huddles of log and sod dwellings. The clumsy vehicle in which they sat was slowing until it was barely crawling between two rows of brick-fronted cabins. You are not lying, Ayna said. I probed your mind, Devin Orth. You are not an android. And I believe that your space ship has been seized by Ivath. It was an alien object on the vast canvas of his pictured world. Here's Hardpan, Red Moustache said, leering slack-jawed at Ayna. Sorry you couldn't go along to New Yok, he added to Orth, you
- 71. and your squirt. She's some fowl. Orth choked and gulped twice. He thanked the driver and climbed down the ladder. Horgan was already standing in the shadow of a doorway above which swung a dust-grimed sign. Two Drik Tony's, the sign read. Wait for us in that store, said Horgan, pointing out a door across the street that was flanked by barrels of fruit and other produce. Orth and me needs a drink. Orth started to protest and then desisted as he saw the girl's eyelid twitch and her head motion toward the door. He followed the frontiersman. Ayna was talking softly to herself as they left her. They joined the men bellied up to the bar. Dun Horgan ordered two shots of alcohol which were brought to them in shallow saucers of glass. Horgan dropped three bits on the bar. How about a shot of tea? he whispered to the bartender. The man's flabby pink face whitened. Imperceptibly he nodded toward the back room and scooped up the three shining coins. The two men downed their fiery drinks and then elbowed their way toward the closed door. It's this accursed Volsad Law, said Horgan. All a result of the Boston tea runners. Tried to smuggle it in and then the reform crowd took it up. Blamed tea for crime and poverty. Pushed the laws through outlawing its sale. Orth grinned. So now the bootleggers, or tealeggers, maybe, are getting rich. Horgan nodded. Inside the door the bartender met them and slipped a small bottle of cold tea into Horgan's pocket. Then he motioned toward the half-open door leading into the alley beyond.
- 72. Please, he said. There may be jeep men watching my bar. They quitted the building and leaving the alley reached the main street. Ayna was waiting in the store's door and as she saw them she started to walk in their direction. A bony stoop-shouldered man with a naked skull beneath his droopy-brimmed hat lurched into her path. His sunken dark eyes were bloodshot and hot. He jerked her arm. Looking for someone? he demanded. I'm here. Ayna's fist landed flush on the man's jaw. He staggered back, but still gripped her. Orth seized the man's shoulder and spun him about. With the same movement his other fist crashed the bony man backward for several paces. But he had not been alone. With him were three other hard-faced men. They helped him to his feet and came pacing toward Orth and Horgan. Their hands were inching down toward their big holstered spring guns. Orth reached for his own hand machine gun, and with his movement their four enemies went for their own weapons. Horgan was slapping his bolts at the quartet. Ayna was hugging the dirty street. Orth felt one smashing impact before his weapon started sewing the explosive little pellets across the four men's middles. Pain was just starting to throb in his left elbow when the last of the others slumped, dead, into the dusty street. Horgan staggered toward him, a six gun bolt in his right side. Just nicked me, he said calmly, his hand holding back the blood that seeped through his coarse-woven shirt. Orth found it hard to believe that these fallen men were actually but pseudo-men, robots. Their laboratory-given life blood was as red and sticky as a true man's, and their dying struggles were as realistic as his own might have been.
- 73. The bartender came sidling up to Orth. He was but one of a score of muttering, staring onlookers. Better clear outta town, he advised. Krepp's brother is sheriff. And if he don't hang you Krepp's mob will do you up. Thanks, Orth said. There were a dozen horses, saddled and bridled, drooping at a nearby hitchrail, and toward these he moved. Come on, he told Horgan and Ayna. We're riding out of here. Horgan shrugged. Might as well get neckties for rustling a horse as for killing Krepp, he conceded, reloading his two spring guns. They climbed into the saddles, Orth snapping a warning burst of explosive slugs into the road and Horgan menacing the glowering knot of townspeople and riders, and went riding eastward out of the village street. Once they were free of the town and climbing a long easy grade into the low tree-clad hills the men of Hardpan City organized their pursuit. Orth saw horses, light waggons, and high-wheeled vehicles resembling bicycles come streaming up the highway after them. Drums began to boom all along the cleared valley they had left and in the hills ahead. News broadcasters, Horgan informed him, warning all cruising scout waggons and squad carts of our escape. Their squad carts are fast—they have pulley drives that can be shifted. If we can only reach the forests again.... We'll make it, Orth said. He grinned encouragingly at Ayna. Maybe we'll find your precious Ivath, too, he added. At that moment they were riding up a short grade, tree-lined and stony, beyond which they could see nothing but an endless stretch of undulating tree-tops. Nothing, Orth was thinking, could now keep them from achieving safety. Suddenly the ground swayed underfoot and their horses spilled them from the saddles.
- 74. There was a moment of rushing blackness, as though they were falling into a pit of tar, and then they felt themselves being whirled horizontally along for a time into a blurring twilight, only to slide softly to a stop. Orth heard a click and a whir from somewhere above him and saw a vast square section of grayness detach itself from the sky above and disappear. He lay quietly for a long minute but the ground was solid underfoot and so he stood up. That, said Ayna, laughing rather breathlessly, was some of Ivath's work. He's brought this section of the crust inside for repairs. She hesitated. Or perhaps because of you, Devin Orth. Me? I get it. If he took the Time Bubble this same way.... Yeah. Orth swallowed thickly. No telling what the mysterious Ivath might be planning to do with them. He was glad Ayna was along. She knew this insane future world. Here he comes now, said Ayna, low-voiced. Ivath, I mean. And, by the way, he is my great grandfather. So don't mind him too much. Orth found himself looking at a transparent bubble of plastic, with a puffy over-sized belt of jade-green metal fixed about its middle. It floated a few feet above the ground, sparks buzzing faintly as it dropped too low and was forced upward again. Inside there was a bony little parody of a man's body, or rather, its upper torso. Below the arms there was nothing save a shining metallic cylinder. The huge blue-veined skull was supported by soft wide bands of plastic material, and the bony arms rested on cushioned ledges. Greetings, Earthman, something inside his brain seemed to say. I have your fellows here, my honored guests. You will join them.
- 75. They are here, my companions? asked Orth stupidly. You mean Horn and Neilson? Did you say that to me? He speaks only in thoughts, said Ayna. When our people reach the age of two hundred they submit to this operation. With their lungs gone there is, of course, no vocal speech. But we live on for centuries untroubled by bodily breakdowns. Ivath motioned with his feeble old arms. Come, he flashed at them, we will join them. As they sat in a small spacer cruising within the vast hollow of Ivath's world-sized stage, Ayna explained more of the mysteries of this future world. How the planets had been cut up into smaller spheres and moved into the dwindling radiations of Sol. How their fleets of space ships crossed the void to trade and mine the precious elements they required, and of the other galactic cultures they met. It is sad, said the girl at last, that you can never return to the past. It is there that our science has utterly failed. Travel in time is but a one-way voyage. You mean, Ayna, Orth said slowly, we can't carry back the knowledge of an atomic shield that will arrest the spread of sterility —that mankind must abandon his use of atomic power? You cannot go back, smiled Ayna, putting her hand on his shoulder as she spoke. But there is no need. In 1980—if our records are not too wrong—Eric Ensamoff discovered such a shield. Great! cried Orth. I won't mind being stranded here. There's Ivath to set right on his ancient history. There's your perfected civilization to study. He swallowed his tongue momentarily and recovered it. And then there's you, Ayna, he blurted. You're....
- 76. The girl slid her fingers across a toggle-switch in the wall. No use letting all the worlds hear us, she said softly, much less see us. You see, I was sent to interview you and get your reactions. All the world was watching while you explored. Orth took the girl and pulled her closer. He studied her face. She smiled. Sure it's turned off? he demanded. She nodded. Fine ... no, they don't need to see this reaction....
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