![Preface to the first edition
This book in based on the courses of lectures “Structurally Stable Systems of Differ-
ential Equations” and “Spaces of Dynamical Systems” given by the author during the
last 30 years to students of the Faculty of Mathematics and Mechanics, St.-Petersburg
State University, specializing in differential equations, geometry, and topology.
As its title indicates, the book is devoted to the theory of dynamical systems (to be
exact, to the structure of spaces of dynamical systems with various topologies).
The world mathematical literature contains a lot of books devoted to dynamical
systems. First we must mention the classical book by Birkhoff [1].
The new approaches to the theory of dynamical systems related to the problem of
structural stability were addressed in the monograph by Nitecki [2].
Later, books devoted to dynamical systems were published by Guckenheimer,
Moser and Newhouse [3], Palis and di Melo [4], Shub [5], Robinson [6], and other
mathematicians.
Finally, let us mention the recent book by Brin and Stuck [7] and the encyclopedic
monograph by Katok and Hasselblatt [8].
In contrast to most of the above-mentioned monographs, the present book is
meant not only for professional mathematicians but also to those who wish to start
the study of dynamical systems, especially students and people working with appli-
cations of the theory of dynamical systems.
Thus, the main goal of the book is to describe the basic objects of the modern
theory of dynamical systems and to formulate its main results.
The first author’s book [9] published in Russian in 1988 served the same purpose.
Comparing the book [9] with the present text, the attentive reader will see that the new
book does not duplicate the old one; in a sense, they are complementary.
In this book, we mostly work with discrete dynamical systems (and not with flows
as in [9]); we describe different approaches to such basic objects as topologies on the
considered spaces of dynamical systems and give principally different proofs of some
basic results, such as structural stability of Anosov diffeomorphisms. Several impor-
tant examples of dynamical systems not included in the book [9] are treated in this
book; let us mention the Bernoulli shift on the space of two-sided sequences, the hy-
perbolic toral automorphism, and the Smale horseshoe.
In addition to well-known fields of dynamical systems (such as topological dy-
namics, theory of structural stability, and chaotic dynamics), the present book con-
tains chapters devoted to C0
-generic properties and shadowing of pseudotrajectories
(the author’s monographs [10, 11] published in the Springer Lect. Notes in Math. series,
vols. 1571 and 1706, were the first monographs in the world mathematical literature de-
voted to these topics).
The book consists of 12 chapters and two appendices.
https://doi.org/10.1515/9783110657166-201](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-13-320.jpg)
![1.1 Main definitions | 3
3. The points f l
(x) and fm
(x) are different if l ̸
= m. In this case, the trajectory of x
is a countable set.
Denote by Per(f) the set of periodic points of a homeomorphism f (we include
fixed points in this set).
Lemma 1.2. The set O(x, f) is finite if and only if x ∈ Per(f).
Proof. It was mentioned above that if x ∈ Per(f ), then the set O(x, f ) is finite.
Let us assume that the set O(x, f ) is finite. In this case, there exist different inte-
ger numbers k and l such that f k
(x) = f l
(x). Let l > k; set n = l − k. Applying the
homeomorphism f −k
to the equality fk
(x) = fl
(x), we see that x = f n
(x). If the points
x, f(x), . . . , f n−1
(x) are distinct, then x is a periodic point of period n.
Otherwise, either x is a fixed point of f or there exist different integer numbers
k1, l1 ∈ [0, n − 1] such that f k1
(x) = fl1
(x). Let k1 < l1 and let n1 = l1 − k1. Then fn1
(y) = y,
where y = fk1
(x) and n1 < n.
Repeating this reasoning, we find a periodic point belonging to O(x, f); clearly, in
this case, x is a periodic point of f.
Now we introduce one more basic notion of theory of dynamical systems. We say
that a set I ⊂ M is invariant for the dynamical system generated by a homeomorphism
f if O(x, f) ⊂ I for any point x ∈ I.
Lemma 1.3. A set I is invariant if and only if f(I) = I.
Proof. Let I be an invariant set. Fix a point x ∈ I. Since O(x, f) ⊂ I, f (x) ∈ I and
f−1
(x) ∈ I. Hence, f(I) ⊂ I and f −1
(I) ⊂ I (thus, I ⊂ f(I)); it follows that f (I) = I.
Inverting the reasoning above, we see that if f(I) = I, then the set I is invariant.
It follows from the well-known properties of homeomorphisms that if I and J are
invariant sets, then the sets I ∪ J, I ∩ J, I J, Cl I, and 𝜕I are invariant as well.
We give an important example of a dynamical system (we refer to this example
below several times).
Example 1.1. Let 𝒳 be the space whose elements are two-sided, infinite, binary se-
quences
a = {ai : ai ∈ {0, 1}, i ∈ ℤ}.
We introduce the following metric in the space 𝒳: If a = {ai} and b = {bi}, we set
dist(a, b) =
∞
∑
i=−∞
|ai − bi|
2|i|
(check that the above formula defines a metric).
Clearly, our definition of the metric dist implies the following statement. For any
given ϵ > 0 there exist numbers N(ϵ) and n(ϵ) such that if
ai = bi, |i| ≤ N(ϵ),](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-23-320.jpg)
![4 | 1 Dynamical systems
then dist(a, b) < ϵ, and if dist(a, b) < ϵ, then
ai = bi, |i| ≤ n(ϵ).
Obviously, N(ϵ), n(ϵ) → ∞ as ϵ → 0.
Let us recall the definition of the Tikhonov product topology (for the particular
case of the space 𝒳) (see [12]). Fix an element a = {ai} of 𝒳 and a finite subset K ⊂ ℤ.
Consider the cylinder
C(a, K) = {b ∈ 𝒳 : bi = ai, i ∈ K}.
Recall that a base of neighborhoods of a point x of a topological space is a family
of neighborhoods of x such that any neighborhood of x contains a neighborhood from
this family.
The base of neighborhoods of a in the Tikhonov product topology consists of the
cylinders C(a, K) corresponding to all finite subsets K ⊂ ℤ.
It is easy to show that the metric topology induced by our metric dist coincides
with the Tikhonov product topology (which means that the families of open sets in
these topologies are the same).
To prove this statement, it is enough to show that if b ∈ C(a, K) for some a ∈ 𝒳
and a finite set K ⊂ ℤ, then there is a small d > 0 such that the metric ball
N(d, b) = {b
∈ 𝒳 : dist(b
, b) < d}
is a subset of C(a, K) and, conversely, if b ∈ N(d, a) for some d > 0, then there is a finite
set K ⊂ ℤ such that C(b, K) ⊂ N(d, a) (we leave details to the reader).
The metric space (𝒳, dist) is compact. This fact follows from the Tikhonov theo-
rem [12] since the space 𝒳 is the countable product of compact spaces {0, 1} and, as
was said, our metric dist induces on 𝒳 the Tikhonov product topology.
Let us give an independent simple proof of the compactness of (𝒳, dist). It is
known that a metric space is compact if and only if any sequence contains a conver-
gent subsequence.
Consider an arbitrary sequence am
= {am
i : i ∈ ℤ}, m ≥ 0.
The elements am
0 take values 0 and 1; hence, there exists a subsequence
m(0) = {m(0, 1), m(0, 2), . . . }
of {0, 1, 2, . . . } such that 0 < m(0, 1) < m(0, 2) < ⋅ ⋅ ⋅ and
am(0,1)
0 = am(0,2)
0 = ⋅ ⋅ ⋅ .
Similarly, there exists a subsequence
m(1) = {m(1, 1), m(1, 2), . . . }](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-24-320.jpg)
![10 | 1 Dynamical systems
smooth vector fields, we formulate analogs of results established for smooth dynam-
ical systems with discrete time. For this reason, we do not give here exact definitions
from the theory of smooth vector fields on manifolds (the reader can find them, for
example, in the book [13]).
Let us briefly recall that a smooth tangent vector field F on a smooth manifold M
is a smooth mapping of the manifold M into its tangent bundle TM.
A smooth curve
γ = ϕ(⋅, x) : I → M,
where I is an interval of the real line, is called the trajectory of a point x ∈ M for the
field F if ϕ(0, x) = x and the tangent vector of γ at the point ϕ(t, x) coincides with the
vector F(ϕ(t, x)) for any t ∈ I.
If the manifold M is compact, then any trajectory of a field F can be continued to
ℝ; thus, in this case any vector field F generates a flow on M.
Let us indicate several relations between the objects defined above and theories
of vector fields and differential equations.
1.2 Embedding of a discrete dynamical system into a flow
Let ϕ be a flow on a topological space M. Fix T > 0 and consider the mapping f : M →
M defined by the formula f (x) = ϕ(T, x).
Let us show that f is a homeomorphism of M. Indeed, if g(x) = ϕ(−T, x), then g is
continuous (see property (CDS3)), and
g(f(x)) = ϕ(−T, ϕ(T, x)) = ϕ(0, x) = x
by properties (CDS2) and (CDS1). A similar reasoning shows that f(g(x)) = x. Hence, g
is the inverse of f, and f is a homeomorphism.
In this case, we say that the homeomorphism f is embedded into the flow ϕ.
If M is a smooth manifold and the flow ϕ is smooth (which means that the map-
pings ϕ(t, ⋅) are smooth for any t), then the same reasoning as above shows that any
homeomorphism f embedded into the flow ϕ is a diffeomorphism.
Clearly, the structure of the set of trajectories of a flow and the structure of the set
of trajectories of a homeomorphism embedded into this flow are closely related.
Nevertheless, one must remember that, in general, properties of the correspond-
ing dynamical systems may differ significantly.
Let us consider the following example.
Example 1.2. Let S be the circle of unit length; introduce on S coordinate x ∈ [0, 1).
Define a flow on S by the equality
ϕ(t, x) = x + t (mod 1);
in this flow, every point moves along the circle into positive direction with unit speed.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-30-320.jpg)
![2 Topologies on spaces of dynamical systems
2.1 C0
-topology
Let (M, dist) be a compact metric space. If f and g are two homeomorphisms of the
space M, we set
ρ0(f, g) = max
x∈M
max(dist(f (x), g(x)), dist(f−1
(x), g−1
(x))). (7)
It is easy to show that ρ0 is a metric on the space of homeomorphisms of the space M.
We denote by H(M) the space of homeomorphisms of the space M with the met-
ric ρ0; the topology induced by the metric ρ0 is called the C0
-topology.
Lemma 2.1. The metric space H(M) is complete.
Proof. Consider a sequence of homeomorphisms fm that is fundamental with respect
to ρ0.
This means that for any ϵ > 0 we can find an index m0 such that ρ0(fl, fk) < ϵ for
k, l > m0.
Then
max
x∈M
dist(fl(x), fk(x)) < ϵ
and
max
x∈M
dist(f−1
l (x), f−1
k (x)) < ϵ
for k, l > m0.
Thus, the sequences fm and f−1
m are fundamental with respect to the uniform metric
r(f, g) = max
x∈M
dist(f (x), g(x)).
Since the space of continuous mappings is complete with respect to the uniform
metric r, there exist continuous mappings f and g of the space M such that r(fm, f) → 0
and r(f−1
m , g) → 0 as m → ∞.
Fix a point x ∈ M. Passing to the limit as m → ∞ in the equality fm(f −1
m (x)) = x, we
see that f (g(x)) = x. Similarly, g(f(x)) = x.
Thus, f is a homeomorphism of the space M, and g = f−1
. Clearly, ρ0(fm, f) → 0 as
m → ∞. The proof is complete.
Remark. It is easy to show that if we replace the metric ρ0 by the uniform metric r,
then the appearing space of homeomorphisms will not be complete.
Indeed, let M be the segment [0, 1]. Fix an integer m > 1 and consider a continuous
mapping fm : [0, 1] → [0, 1] defined as follows: fm(0) = 0, fm(1) = 1, fm(1/3) = 1/m,
https://doi.org/10.1515/9783110657166-002](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-36-320.jpg)
![2.2 C1
-topology | 17
fm(2/3) = 1−1/m, and fm is affine on any of the segments [0, 1/3], [1/3, 2/3], and [2/3, 1].
Clearly, fm is a homeomorphism of the segment [0, 1].
It is easily seen that the inequality
fm(x) − fn(x)
≤ max(
1
m
,
1
n
), x ∈ [0, 1],
holds for any m, n > 1; thus, the sequence fm is fundamental with respect to the uni-
form metric r. Hence, this sequence converges with respect to r, and the limit function
f equals 0 on [0, 1/3] and 1 on [2/3, 1]. Hence, f is not a homeomorphism.
Now let ϕ and ψ be two flows on a compact metric space (M, dist). It was shown
in Section 1.2 that, for any t ̸
= 0, the mapping ϕ(t, ⋅) is a homeomorphism of the space
M (for t = 0, the mapping ϕ(0, ⋅) = Id is a homeomorphism as well). Define
ρ0(ϕ, ψ) = max
t∈[−1,1]
ρ0(ϕ(t, ⋅), ψ(t, ⋅)). (8)
Clearly, ρ0 is a metric on the space of flows on M. We denote by ℱ0
(M) the space of
flows on M with the metric ρ0; similarly to the case of the space of homeomorphisms,
the topology induced by the metric ρ0 is called the C0
-topology.
The same reasoning as in the proof of Lemma 2.1 shows that ℱ0
(M) is a complete
metric space.
2.2 C1
-topology
Let M be a smooth closed (i. e., compact and boundaryless) manifold. To introduce the
C1
-topology on the space of diffeomorphisms of M, we assume that M is a submanifold
of the Euclidean space ℝN
(a different, equivalent, approach to the definition of the
C1
-topology based on local coordinates is described in [9]).
No generality is lost assuming that M is a submanifold of a Euclidean space since,
by the classical Whitney theorem, any smooth closed manifold can be embedded into
a Euclidean space of appropriate dimension.
If M is a submanifold of ℝN
, for any point x ∈ M we can identify the tangent space
TxM of M at x with a linear subspace of ℝN
. Consider the metric dist on M induced by
the Euclidean metric of the space ℝN
. For a vector v ∈ TxM we denote by |v| its norm
as the norm in the space ℝN
.
Let f and g be two diffeomorphisms of the manifold M. Define the value ρ0(f, g)
by the same formula (7) as for homeomorphisms of a compact metric space.
Take a point x of the manifold M and a vector v from the tangent space TxM. We
consider the vectors Df (x)v ∈ Tf(x)M and Dg(x)v ∈ Tg(x)M as vectors of the same Eu-
clidean space ℝN
. Hence, the following values are defined: |Df (x)v − Dg(x)v| and
Df(x) − Dg(x)
= sup
v∈TxM,|v|=1
Df(x)v − Dg(x)v
.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-37-320.jpg)
![18 | 2 Topologies on spaces of dynamical systems
Similarly, one defines the value
Df−1
(x) − Dg−1
(x)
= sup
v∈TxM,|v|=1
Df−1
(x)v − Dg−1
(x)v
.
Introduce the number
ρ1(f , g) = ρ0(f, g) + sup
x∈M
Df(x) − Dg(x)
+ sup
x∈M
Df −1
(x) − Dg−1
(x)
.
Clearly, ρ1 is a metric on the space of diffeomorphisms of the manifold M. We denote
by Diff1
(M) the space of diffeomorphisms of M with metric ρ1; the topology induced
by the metric ρ1 is called the C1
-topology.
The standard reasoning (left to the reader) shows that (Diff1
(M), ρ1) is a complete
metric space.
Now we consider the space of smooth flows on M. We say that a flow ϕ : ℝ×M → M
is smooth if for any t ∈ ℝ, the mapping ϕ(t, ⋅) is smooth (for our purposes, it is enough
to assume that this mapping is of class C1
; this assumption is satisfied if we consider
a flow generated by a vector field of class C1
).
Our reasoning above (see Section 1.2) shows that if a flow ϕ : ℝ × M → M is
smooth, then the mapping ϕ(t, ⋅) is a diffeomorphism of the manifold M for any t.
If ϕ and ψ are two smooth flows on M, we set
ρ1(ϕ, ψ) = max
t∈[−1,1]
ρ1(ϕ(t, ⋅), ψ(t, ⋅)). (9)
It is easy to show that ρ1 is a metric on the space of smooth flows on M; we denote
by ℱ1
(M) the space of smooth flows on M with the metric ρ1. Similarly to the case of
diffeomorphisms, the topology induced by the metric ρ1 is called the C1
-topology.
2.3 Metrics on the space of systems of differential equations
Considering flows generated by vector fields on smooth closed manifolds, we have
introduced two metrics, ρ0 and ρ1. Defining these metrics, we estimated differences
between trajectories of the flows with the same initial values and between the corre-
sponding “variational flows” on time intervals of fixed length.
Considering flows generated by autonomous systems of differential equations, it
is natural to study metrics that are based on differences between right-hand sides of
the systems rather than on differences between trajectories.
As was mentioned in Section 1.1, we denote by |x| the Euclidean norm of a vector
x ∈ ℝn
.
For an n × n matrix A, let us denote by ‖A‖ its operator norm generated by | ⋅ |, i. e.,
the value
‖A‖ = max
|x|=1
|Ax|.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-38-320.jpg)
![20 | 2 Topologies on spaces of dynamical systems
and
r1(F, G) = r0(F, G) + sup
x∈ℝn
𝜕F
𝜕x
(x) −
𝜕G
𝜕x
(x)
are finite (in fact, we study the case where these values tend to 0).
Below, we refer to the following elementary estimate of the difference between
solutions of two systems of differential equations (to prove this estimate, it is enough
to represent the considered solutions of Cauchy problems in the form of solutions of
equivalent integral equations and to apply the Gronwall lemma; a similar but more
complicated estimate is applied in the following proof of Lemma 2.2).
Consider two systems of differential equations,
dx
dt
= f (t, x) (13)
and
dx
dt
= g(t, x), (14)
where x ∈ ℝn
.
Assume that the vector-functions f and g are continuous in ℝn+1
, f is globally Lip-
schitz continuous in x with constant L, and the value
m = sup
(t,x)∈ℝn+1
f (t, x) − g(t, x)
is finite.
If x(t) and y(t) are solutions of systems (13) and (14), respectively, that are defined
on the same segment [a, b] and have the same initial values (t0, x0), where t0 ∈ [a, b],
then
x(t) − y(t)
≤ m exp(L(b − a)), t ∈ [a, b]. (15)
Lemma 2.2.
(1) If r0(F, G) → 0, then ρ0(ϕ, ψ) → 0.
(2) If r1(F, G) → 0, then ρ1(ϕ, ψ) → 0.
Proof. Let x be an arbitrary point of the space ℝn
. Since ϕ(t, x) and ψ(t, x) are solutions
of systems (11) and (12), respectively, with the same initial values (0, x), estimate (15)
implies that
ρ0(ϕ, ψ) = sup
x∈ℝn
max
t∈[−1,1]
ϕ(t, x) − ψ(t, x)
≤ r0(F, G) exp(L).
This proves statement (1).](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-40-320.jpg)
![2.3 Metrics on the space of systems of differential equations | 21
To estimate the value ρ1(ϕ, ψ), we again fix a point x ∈ ℝn
and consider the deriva-
tives of the flows ϕ and ψ with respect to initial values,
Y(t) =
𝜕ϕ
𝜕x
(t)
and
Z(t) =
𝜕ψ
𝜕x
(t).
Recall that these derivatives are matrix-valued solutions of the variational systems
dY
dt
= Φ(t, Y), where Φ(t, Y) =
𝜕F
𝜕x
(t, ϕ(t))Y, (16)
and
dZ
dt
= Ψ(t, Z), where Ψ(t, Z) =
𝜕G
𝜕x
(t, ψ(t))Z, (17)
respectively, with the same initial values Y(0) = Z(0) = E, where E is the identity
matrix of size n × n.
It was assumed that the Jacobi matrix 𝜕F/𝜕x is bounded; in addition, since
r1(F, G) → 0, we may assume, for example, that r1(F, G) ≤ 1. Hence, there exists a
number N > 0 such that
𝜕F
𝜕x
≤ N,
𝜕G
𝜕x
≤ N, x ∈ ℝn
. (18)
First we estimate the value ‖Z(t)‖ for t ∈ [−1, 1] (we are not going to get an exact
estimate of the value ‖Z(t)‖; it is important for us to estimate this value by a constant
depending on N and independent from the initial point x of the trajectory ψ(t, x)).
If z is a column of the matrix Z, then
dz
dt
=
𝜕G
𝜕x
(t, ψ(t))z (19)
and |z(0)| = 1.
We take the scalar product of equality (19) and the vector z(t):
⟨
dz
dt
, z⟩ = ⟨
𝜕G
𝜕x
(t, ψ(t)), z⟩, (20)
where ⟨⋅⟩ denotes scalar product.
As above, we denote by z2
(t) the square of the Euclidean norm of z(t). Relations
(18) and (20) imply that
1
2
d
dt
z2
= ⟨
dz
dt
, z⟩ = ⟨
𝜕G
𝜕x
(t, ψ(t))z, z⟩
≤
𝜕G
𝜕x
(t, ψ(t))z
|z| ≤
𝜕G
𝜕x
(t, ψ(t))
|z||z| ≤ Nz2
.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-41-320.jpg)
![2.4 Generic properties | 23
Relations (22) and (23) imply that
Y(t) − Z(t)
≤
t
∫
0
Φ(s, Y(s)) − Ψ(s, Z(s))
ds
.
The integrand is estimated as follows:
Φ(s, Y(s)) − Ψ(s, Z(s))
≤
Φ(s, Y(s)) − Φ(s, Z(s))
+
Φ(s, Z(s)) − Ψ(s, Z(s))
.
It follows from estimates (18) that
Φ(s, Y(s)) − Φ(s, Z(s))
=
𝜕F
𝜕x
(ϕ(s, x))(Y(s) − Z(s))
≤ N
Y(s) − Z(s)
. (24)
Further, if r1(F, G) < δ1, then
Φ(s, Z(s)) − Ψ(s, Z(s))
≤
(
𝜕F
𝜕x
(ϕ(s, x)) −
𝜕F
𝜕x
(ψ(s, x)))Z(s)
+
(
𝜕F
𝜕x
(ψ(s, x)) −
𝜕G
𝜕x
(ψ(s, x)))Z(s)
≤ (ϵ + r1(F, G))N2.
Combining these inequalities with estimate (24), we get the inequality
Y(t) − Z(t)
≤
t
∫
0
(N
Y(s) − Z(s)
+ (ϵ + r1(F, G))N2) ds
.
Applying the Gronwall lemma to the above inequality, we conclude that if
r1(F, G) < δ1, then
Y(t) − Z(t)
≤ (ϵ + r1(F, G))N2 exp(N)
for t ∈ [−1, 1].
Since ϵ is arbitrary, the last inequality implies statement (2) of our lemma.
2.4 Generic properties
In the global theory of dynamical systems, it is important to study generic properties.
Let X be a topological space. We say that a set A ⊂ X is residual if there exists a
countable family of open subsets {An : n ∈ ℤ} of the space X such that
⋂
n∈ℤ
An = A. (25)
A property of elements of the space X is called generic if there exists a residual
subset of X such that every element of this subset has this property. If a property is
generic, we say that a generic element of the space X has this property.
The following classical theorem was proven by Baire.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-43-320.jpg)
![24 | 2 Topologies on spaces of dynamical systems
Theorem 2.1. If X is a complete metric space, then any residual subset is dense in X.
A residual set is “large” in the topological sense. At the same time, a residual set
can be “small” in the sense of measure.
Let us show that there exists a residual subset of the real line ℝ whose Lebesgue
measure is zero. Fix a countable dense subset {an, n = 0, 1, . . . } of the line (for example,
consider the set of rational numbers).
Take a natural number m and consider the set
Am = ⋃
n≥0
(an −
1
m2n
, an +
1
m2n
).
The set Am is an open and dense subset of the line, and its Lebesgue measure can be
estimated as follows:
mes Am ≤
∞
∑
n=0
2
m2n
=
4
m
.
The set A = ∩m>0Am is residual, and mes A = 0.
It was mentioned in Section 1.2 that there exist diffeomorphisms that cannot be
embedded into flows. The precise statement of the main result of [14] is as follows.
Theorem 2.2. Let M be a smooth closed manifold whose dimension is not less than 2.
Then a generic diffeomorphism in Diff1
(M) cannot be embedded into a flow generated
by a Lipschitz continuous vector field on M.
2.5 Immersions and embeddings
In this book, we consider two main classes of mappings studied in topology, immer-
sions and embeddings. We mostly restrict ourselves to immersions and embeddings of
smooth manifolds (Euclidean spaces and disks in such spaces) into Euclidean spaces.
Let us recall the basic definitions.
Let X and Y be topological spaces. A mapping f from X to Y is a topological im-
mersion if any point x ∈ X has a neighborhood U such that f maps homeomorphically
U onto f (U).
Now let f be a mapping of a smooth manifold M to a smooth manifold N.
We say that f is an immersion of class Ck
, k ≥ 1, if f belongs to class Ck
and
rank Df (x) = dim M
for any x ∈ M (let us note that the last condition implies that dim N ≥ dim M).
The mapping f is called an embedding of class Ck
, k ≥ 1, if f (M) is a submanifold
of N and f is a diffeomorphism of M and f(M).
We will work with smooth disks. By definition, a smooth disk is the image of a ball
of a Euclidean space under an embedding.](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-44-320.jpg)
![3.1 Topological conjugacy | 27
For example, periodic trajectories of the homeomorphism f are mapped to peri-
odic trajectories of the homeomorphism g. Indeed, let p be a periodic point of f of
period m, i. e., the points
p0 = p, p1 = f(p), . . . , pm−1 = fm−1
(p)
are distinct and fm
(p) = p. If r = h(p), then
ri = gi
(r) = gi
(h(p)) = h(f i
(p)) = h(pi)
by Lemma 3.1; thus, the points ri, i = 0, . . . , m − 1, are distinct and gm
(r) = h(f m
(p)) =
h(p) = r.
A similar reasoning shows that if a trajectory O(p, f ) (a semitrajectory O+
(p, f)) is
dense in M, then the trajectory O(h(p), g) (semitrajectory O+
(h(p), g)) is dense in N.
Remark. Sometimes, it is possible to significantly simplify a problem by passing from
a homeomorphism to a topologically conjugate homeomorphism (this idea will be ap-
plied in Chapter 9, where we study the Smale horseshoe).
Here we consider as an example two semidynamical systems.
Let f be a continuous mapping of a topological space M into itself. We set ϕ(m, x) =
fm
(x), m ∈ ℤ+, and get a semidynamical system, i. e., a mapping ϕ : ℤ+ × M → M
whose properties are similar to properties (DDS1)–(DDS3) (one has to replace ℤ by ℤ+
in properties (DDS2) and (DDS3)).
The trajectory of a point x in the semidynamical system ϕ is defined by the equality
O(x, ϕ) = {ϕ(m, x) : m ∈ ℤ+};
the definition of a periodic point is literally the same as in the case of a dynamical
system.
Two semidynamical systems generated by mappings f and g are called topologi-
cally conjugate if there exists a homeomorphism h that satisfies equality (26).
Exercise 3.1. Consider two semidynamical systems on the segment [0, 1] generated by
the mappings
f(x) = 4x(1 − x)
and
g(x) = {
2x, x ∈ [0, 1/2],
2(1 − x), x ∈ (1/2, 1].
Prove that the mapping
h(x) =
2
π
arcsin √x](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-47-320.jpg)
![28 | 3 Equivalence relations
is a homeomorphism of the segment [0, 1] that conjugates the semidynamical systems
generated by the mappings f and g.
Thus, we can reduce the study of the dynamics of the essentially nonlinear map-
ping f to the similar problem for the piecewise linear mapping g.
In what follows, we consider dynamical systems with the same phase space M.
Lemma 3.2. Topological conjugacy is an equivalence relation on the space H(M).
Proof. Since the identity homeomorphism Id conjugates any homeomorphism with
itself, topological conjugacy is reflexive.
Topological conjugacy is symmetric. Indeed, if a homeomorphism h conjugates
f and g, i. e., g ∘ h = h ∘ f, then h−1
conjugates g and f ; to show this, apply h−1
to the
equality g∘h = h∘f both from the right and left. As a result, we get the desired equality
f ∘ h−1
= h−1
∘ g.
Finally, we show that topological conjugacy is transitive. Assume that h1 conju-
gates f and g and h2 conjugates g and k. Then h = h2 ∘ h1 is a homeomorphism of the
space M, and
h ∘ f = h2 ∘ (h1 ∘ f) = h2 ∘ (g ∘ h1) = (k ∘ h2) ∘ h1 = k ∘ h,
which completes the proof of our lemma.
Of course, topological conjugacy is an equivalence relation on the space of diffeo-
morphisms Diff1
(M) of a smooth manifold M as well.
This relation allows us to give the main definition of the theory of structural sta-
bility.
Let M be a smooth closed manifold. A diffeomorphism f ∈ Diff1
(M) is called
structurally stable if there exists a neighborhood W of the diffeomorphism f in the
C1
-topology such that any diffeomorphism g ∈ W is topologically conjugate with f.
The above definition and Lemma 3.2 imply that any diffeomorphism g ∈ W is
structurally stable as well. Denote by 𝒮(M) the set of structurally stable diffeomor-
phisms in Diff1
(M). Clearly, the following statement holds (since this statement is very
important for us, we call it a theorem).
Theorem 3.1. The set 𝒮(M) is open in Diff1
(M).
The property of structural stability was first defined by Andronov and Pontryagin
for autonomous systems of differential equations (we discuss this definition below, in
Section 7.6).
In fact, this original definition corresponds to a slightly different property which
we formulate below.
A diffeomorphism f ∈ Diff1
(M) is called structurally stable in the strong sense if for
any ϵ > 0 one can find a neighborhood W of the diffeomorphism f in the C1
-topology](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-48-320.jpg)
![enterprise if 'passed' means 'crossed,' as in the 'Grindelwald' Sketch-
Book, against a drawing of mountains, is the following scrawl:—
'No matter what bef [? befell] Hannibel—W.B. and J.M.W.T.
passed the Alps from [? near] Fombey [?] Sep. 3, 1844.'
CHAPTER LI
1845: AGED SEVENTY
PICTURES OF WHALERS, AND AN ENTRY ON THE LAST PAGE
OF HIS LAST SKETCH-BOOK
Now, when he is nearing his decline, Turner is described as stooping
very much, and looking down. Thinking of Turner 'looking down,' I
recall the story that came to Sir Walter Armstrong from Mr. Stopford
Brooke: how some one who knew Turner, at least by sight, was one
day passing along the wharves beyond the Palace of Westminster,
when he noticed the figure of a sturdy man in black squatting on his
heels at the river's edge, and looking down intently into the water.
Passing on, he thought for the moment no more about it. But on his
return, half an hour later, the figure was still there, and still intent in
the same way. That watcher was Turner, and the object of his interest
was the pattern made by the ripples at the edge of the tide.
Ruskin says that this year his health, and with it in great degree, his
mind, failed suddenly. And to Ruskin we owe this pathetic passage:—
'The last drawing in which there remained a reflection of his
expiring power, he made in striving to realise, for me, one of
these faint and fair visions of the morning mist fading from the
Lake of Lucerne.
'"There ariseth a little cloud out of the sea like a man's hand ...
For what is your life?"'](https://image.slidesharecdn.com/25469245-250524054427-51f132ad/85/Spaces-Of-Dynamical-Systems-2-Rev-And-Exten-Edition-Sergei-Yu-Pilyugin-71-320.jpg)
Spaces Of Dynamical Systems 2 Rev And Exten Edition Sergei Yu Pilyugin
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- 5. Sergei Yu. Pilyugin Spaces of Dynamical Systems
- 6. De Gruyter Studies in Mathematical Physics | Edited by Michael Efroimsky, Bethesda, Maryland, USA Leonard Gamberg, Reading, Pennsylvania, USA Dmitry Gitman, São Paulo, Brazil Alexander Lazarian, Madison, Wisconsin, USA Boris Smirnov, Moscow, Russia Volume 3
- 7. Sergei Yu. Pilyugin Spaces of Dynamical Systems | 2nd edition
- 8. Physics and Astronomy Classification 2010 01.30 mm, 02.30 Hq, 05.45.-a, 05.45 Ac Author Prof. Dr. Sergei Yu. Pilyugin St. Petersburg State University Universitetskaya nab 7–9 St. Petersburg 199034 Russia ISBN 978-3-11-064446-3 e-ISBN (PDF) 978-3-11-065716-6 e-ISBN (EPUB) 978-3-11-065399-1 ISSN 2194-3532 Library of Congress Control Number: 2019938959 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2019 Walter de Gruyter GmbH, Berlin/Boston Cover image: Science Photo Library/Heller, Eric Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com
- 9. Contents Preface to the first edition | IX Preface to the second edition | XIII List of main symbols | XV 1 Dynamical systems | 1 1.1 Main definitions | 1 1.2 Embedding of a discrete dynamical system into a flow | 10 1.3 Local Poincaré diffeomorphism | 11 1.4 Time-periodic systems of differential equations | 13 1.5 Action of a group | 14 2 Topologies on spaces of dynamical systems | 16 2.1 C0 -topology | 16 2.2 C1 -topology | 17 2.3 Metrics on the space of systems of differential equations | 18 2.4 Generic properties | 23 2.5 Immersions and embeddings | 24 3 Equivalence relations | 26 3.1 Topological conjugacy | 26 3.2 Topological equivalence of flows | 30 3.3 Nonwandering set | 30 3.4 Local equivalence | 36 4 Hyperbolic fixed point | 37 4.1 Hyperbolic linear mapping | 37 4.2 The Grobman–Hartman theorem | 40 4.3 Neighborhood of a hyperbolic fixed point | 47 4.4 The stable manifold theorem | 52 4.5 Hyperbolic periodic point | 64 5 Hyperbolic rest point and hyperbolic closed trajectory | 66 5.1 Hyperbolic rest point | 66 5.2 Hyperbolic closed trajectory | 71
- 10. VI | Contents 6 Transversality | 77 6.1 Transversality of mappings and submanifolds | 77 6.2 Transversality condition | 79 6.3 Palis lemma | 81 6.4 Transversality and hyperbolicity for one-dimensional mappings | 89 7 Hyperbolic sets | 90 7.1 Definition of a hyperbolic set | 90 7.2 Examples of hyperbolic sets | 92 7.3 Basic properties of hyperbolic sets | 95 7.4 Stable manifold theorem | 99 7.5 Axiom A | 101 7.6 Hyperbolic sets of flows | 109 8 Anosov diffeomorphisms | 116 9 Smale’s horseshoe and chaos | 123 9.1 Smale’s horseshoe | 123 9.2 Chaotic sets | 128 9.3 Homoclinic points | 129 10 Closing lemma | 131 11 C0 -generic properties of dynamical systems | 136 11.1 Hausdorff metric | 136 11.2 Semicontinuous mappings | 137 11.3 Tolerance stability and Takens’ theory | 138 11.4 Attractors of dynamical systems | 142 12 Shadowing of pseudotrajectories in dynamical systems | 153 12.1 Definitions and results | 153 12.2 Proof of Theorem 12.1 | 158 12.3 Proof of Theorem 12.2 | 166 12.4 Proof of Theorem 12.3 | 169 13 Invariant measures | 174 13.1 Main definitions | 174 13.2 Construction of invariant measures | 175 13.3 Krylov–Bogolyubov theorem | 179 13.4 Ergodic theorems | 180 13.5 Poincaré recurrence and Hamiltonian systems | 189 13.6 Ergodic closing lemma | 195
- 11. Contents | VII A Scheme of the proof of the Mañé theorem | 196 B Lectures on selected chapters of the history of differential equations and dynamical systems | 206 B.1 Differential equations and Newton’s anagram | 206 B.2 Development of the general theory | 208 B.3 Linear equations and systems | 212 B.4 Stability | 217 B.5 Nonlocal qualitative theory. Dynamical systems | 224 B.6 Structural stability | 228 B.7 Dynamical systems with chaotic behavior | 231 Bibliography | 236 Index | 239
- 13. Preface to the first edition This book in based on the courses of lectures “Structurally Stable Systems of Differ- ential Equations” and “Spaces of Dynamical Systems” given by the author during the last 30 years to students of the Faculty of Mathematics and Mechanics, St.-Petersburg State University, specializing in differential equations, geometry, and topology. As its title indicates, the book is devoted to the theory of dynamical systems (to be exact, to the structure of spaces of dynamical systems with various topologies). The world mathematical literature contains a lot of books devoted to dynamical systems. First we must mention the classical book by Birkhoff [1]. The new approaches to the theory of dynamical systems related to the problem of structural stability were addressed in the monograph by Nitecki [2]. Later, books devoted to dynamical systems were published by Guckenheimer, Moser and Newhouse [3], Palis and di Melo [4], Shub [5], Robinson [6], and other mathematicians. Finally, let us mention the recent book by Brin and Stuck [7] and the encyclopedic monograph by Katok and Hasselblatt [8]. In contrast to most of the above-mentioned monographs, the present book is meant not only for professional mathematicians but also to those who wish to start the study of dynamical systems, especially students and people working with appli- cations of the theory of dynamical systems. Thus, the main goal of the book is to describe the basic objects of the modern theory of dynamical systems and to formulate its main results. The first author’s book [9] published in Russian in 1988 served the same purpose. Comparing the book [9] with the present text, the attentive reader will see that the new book does not duplicate the old one; in a sense, they are complementary. In this book, we mostly work with discrete dynamical systems (and not with flows as in [9]); we describe different approaches to such basic objects as topologies on the considered spaces of dynamical systems and give principally different proofs of some basic results, such as structural stability of Anosov diffeomorphisms. Several impor- tant examples of dynamical systems not included in the book [9] are treated in this book; let us mention the Bernoulli shift on the space of two-sided sequences, the hy- perbolic toral automorphism, and the Smale horseshoe. In addition to well-known fields of dynamical systems (such as topological dy- namics, theory of structural stability, and chaotic dynamics), the present book con- tains chapters devoted to C0 -generic properties and shadowing of pseudotrajectories (the author’s monographs [10, 11] published in the Springer Lect. Notes in Math. series, vols. 1571 and 1706, were the first monographs in the world mathematical literature de- voted to these topics). The book consists of 12 chapters and two appendices. https://doi.org/10.1515/9783110657166-201
- 14. X | Preface to the first edition In Chapter 1, we define the main objects, dynamical systems with continuous and discrete time. We describe possible types of trajectories and the basic properties of invariant sets. As an example, we consider the Bernoulli shift on the space of two- sided sequences. We study embeddings of discrete dynamical systems into flows and the local Poincaré transformation. In Chapter 2, we introduce the C0 topology on the space of homeomorphisms of a compact metric space and the C1 topology on the space of diffeomorphisms of a smooth closed manifold. For flows generated by autonomous systems of ordinary dif- ferential equations, we describe relations between two possible approaches to defin- ing the topology: via estimates of differences between the right-hand sides of the sys- tems and via estimates of closeness of the flows. We consider Baire spaces and generic properties. In Chapter 3, we study the main equivalence relations on spaces of dynamical sys- tems: topological conjugacy of systems with discrete time and topological equivalence of systems with continuous time. Structural stability and Ω-stability are defined. We introduce the nonwandering set of a dynamical system and prove the Birkhoff theo- rem: Any trajectory lives only a finite time outside a neighborhood of the nonwander- ing set. Chapter 4 is one of the main parts of the book. In this chapter, we define the basic concepts of the theory of structural stability (such as stable and unstable manifolds, fundamental domains, etc.) in the simplest case of a hyperbolic fixed point. We de- scribe properties of hyperbolic linear mappings and prove the Grobman–Hartman the- orem on the local topological conjugacy of a diffeomorphism near its hyperbolic fixed point and the corresponding linear mapping. A detailed proof of the stable manifold theorem is given; the proof is based on the Perron method. The case of a hyperbolic periodic point is considered as well. In Chapter 5, we prove analogs of results obtained in Chapter 4 for the case of rest points and closed trajectories of an autonomous system of differential equations. It is shown how to reformulate the definition of hyperbolicity of a closed trajectory in terms of multiplicators of the corresponding periodic solution. Chapter 6 is devoted to transversality. We define transversality of mappings and submanifolds. The property of transversality of stable and unstable manifolds is in- troduced. We prove the Palis λ-lemma and describe relations between transversality and hyperbolicity for one-dimensional mappings. In Chapter 7, the second main part of the book, we study hyperbolic sets. We ana- lyze the definition of a hyperbolic set and give two basic examples of a hyperbolic set: a hyperbolic fixed point and a hyperbolic automorphism of the torus. We formulate the stable manifold theorem, introduce Axiom A, and prove the spectral decomposition theorem. The main results of the theory of structural stability are formulated. Hyper- bolic sets of flows are described. We analyze relations between the structural stability theorem and the classical Andronov–Pontryagin theorem on “roughness” of planar autonomous systems.
- 15. Preface to the first edition | XI In Chapter 8, we prove the structural stability of an Anosov diffeomorphism. Chapter 9 is devoted to Smale’s horseshoe and chaos. We prove that the horse- shoe invariant set is topologically conjugate to the Bernoulli shift. It is shown that the horseshoe invariant set is chaotic. Transverse homoclinic points of planar diffeomor- phisms are considered. We formulate the classical C1 closing lemma in Chapter 10. The C0 closing lemma is proven. In Chapter 11, we study C0 generic properties of dynamical systems. The Hausdorff metric is defined. The main results of Takens theory related to the tolerance stability conjecture are proven. The second part of Chapter 11 is devoted to the behavior of at- tractors under C0 small perturbations. We prove the Hurley theorem on genericity of stability of attractors in the Hausdorff metric under C0 small perturbations. Chapter 12 is devoted to shadowing of pseudotrajectories. We prove that a hyper- bolic set has the Lipschitz shadowing property. The Lipschitz inverse shadowing prop- erty for a trajectory having (C, λ)-structure is established. The proofs of these results are based on the Tikhonov–Schauder fixed point theorem. Shadowing and inverse shadowing properties of linear mappings are completely characterized. In Appendix A, we describe a scheme of the proof of Mañé’s theorem on the ne- cessity of hyperbolicity for structural stability. Appendix B is devoted to the history of the theory of differential equations and dy- namical systems. The sections are concerned with differential equations and Newton’s anagram; development of the general theory; linear equations and systems; stability; nonlocal qualitative theory; dynamical systems; structural stability, and dynamical systems with chaotic behavior. This text is based on lectures on the history of mathe- matics given by the author in the last years to PhD students of the Faculty of Mathe- matics and Mechanics. In the text, we do not give references to basic University mathematical courses. For the author of this book, it was very important to read books and research pa- pers on differential equations and dynamical systems. At the same time, the author is grateful to many mathematicians for personal contacts. First, the author wants to thank his teachers in differential equations, dynami- cal systems, and topology: Yu. N. Bibikov, S. M. Lozinskii, N. N. Petrov, V. A. Pliss, and V. A. Rokhlin. The author is grateful for cooperation with his colleagues at the Faculty of Mathe- matics and Mechanics: L. Ya. Adrianova, A. F. Andreev, V. E. Chernyshev, Yu. V. Churin, Yu. A. Il’in, O. A. Ivanov, S. G. Kryzhevich, G. A. Leonov, N. Yu. Netsvetaev, and A. V. Os- ipov. It was very useful to discuss dynamical systems and related fields of mathe- matics with D. V. Anosov, V. S. Afraimovich, V. I. Arnold, Yu. S. Ilyashenko, V. M. Mil- lionshchikov, Yu. I. Neimark, G. S. Osipenko, N. Kh. Rozov, A. N. Sharkovskii, and L. P. Shilnikov, and also with W.-J. Beyn, B. Fiedler, and P. Kloeden (Germany),
- 16. XII | Preface to the first edition G. R. Sell and J. K. Hale (USA), K. Palmer (Taiwan), L. Wen and S. Gan (China), C. Bon- atti (France), L. Diaz (Brazil), T. Eirola (Finland), R. Corless (Canada), and K. Sakai (Japan). The author is happy to mention several of his students who have contributed to the development of the theory of dynamical systems: N. Ampilova, A. Felshtyn, A. Katina, A. Osipov, O. Plamenevskaya, V. Pogonysheva, O. Tarakanov, and S. Tikhomirov. The English text of the book is slightly modified compared to the Russian one. The structure of chapters and appendices is the same, but some details of presentation are improved, for the convenience of the general reader we give references to books published in the West instead of Russian ones, and so on.
- 17. Preface to the second edition The second edition of the book “Spaces of Dynamical Systems” is an expanded and corrected variant of its first edition published by Walter De Gruyter, Berlin/Boston in 2012 in the series “Studies in Mathematical Physics,” Vol. 3. This edition contains a new chapter, Chapter 13, on invariant measures, in which we describe the construction of invariant measures, prove the Krylov–Bogolyubov the- orem on the existence of an invariant measure for a continuous mapping of a com- pact metric space, the von Neumann and Birkhoff ergodic theorems, and the Poincaré recurrence theorem. We introduce the simplest variant of Hamiltonian systems with phase space ℝ2N and prove the Liouville theorem on the density of the integral invari- ant to show that the flow of a Hamiltonian system preserves volume. In addition, several proofs of the first edition were modified and simplified. The list of references includes several new related publications. During the preparation of the second edition, the author was supported by the P. L. Chebyshev Laboratory, St.Petersburg State University, and by the Russian Foun- dation for Basic Research, grant 18-01-00230. https://doi.org/10.1515/9783110657166-202
- 19. List of main symbols ℝn the Euclidean n-space (we write ℝ instead of ℝ1 ) ℂn the complex n-space (we write ℂ instead of ℂ1 ) ℤ the set of integers ℤ+ the set of nonnegative integers ℤ− the set of nonpositive integers E the identity matrix diag(A1, . . . , Am) a block-diagonal matrix with blocks A1, . . . , Am Id the identity mapping f ∘ g the composition of mappings f and g 𝜕f 𝜕x the partial derivative of a mapping f in variable x Ck (U, V) the class of continuous mappings from U to V having continuous derivatives up to order k Df the derivative of a mapping f TxM the tangent space of a manifold M at a point x dim M the dimension of a manifold M N(a, A) the a-neighborhood of a set A Cl A the closure of a set A Int A the interior of a set A 𝜕A the boundary of a set A card A the cardinality of a finite set A Lp (X, μ) the space of pth power integrable functions on a measure space (X, μ) https://doi.org/10.1515/9783110657166-203
- 21. 1 Dynamical systems 1.1 Main definitions Theory of dynamical systems studies two main classes of dynamical systems, systems with discrete time (cascades) and systems with continuous time (flows). We first define a dynamical system with discrete time. Let f be a homeomorphism of a topological space M. We define (functional) de- grees of f as follows: Set f 0 = Id, where Id is the identical mapping of M; if m is natural, we set f m = f ∘ f ∘ ⋅ ⋅ ⋅ ∘ f ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ m times ; finally, if m is a negative integer, we set f m = f −1 ∘ f−1 ∘ ⋅ ⋅ ⋅ ∘ f −1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ |m| times , where f−1 is the inverse of f. Clearly, the mappings f m are continuous for all m ∈ ℤ. Denote ϕ(m, x) = fm (x). It is easily seen that the mapping ϕ : ℤ × M → M has the following three properties: (DDS1) ϕ(0, x) = x, x ∈ M; (DDS2) ϕ(l + m, x) = ϕ(l, ϕ(m, x)), l, m ∈ ℤ, x ∈ M; (DDS3) for any m ∈ ℤ, the mapping ϕ(m, ⋅) is continuous. Any mapping ϕ : ℤ×M → M having properties (DDS1)–(DDS3) is called a (continuous) dynamical system with discrete time (sometimes such a system is called a cascade). The space M is called the phase space of the system. It is easy to understand that if we are given a mapping ϕ : ℤ × M → M having properties (DDS1)–(DDS3), then there exists a homeomorphism f such that ϕ(m, x) = fm (x). Indeed, set f(x) = ϕ(1, x). Let us show that f is a homeomorphism. The mapping f is continuous by property (DDS3); the mapping g(x) = ϕ(−1, x) is continuous as well, and properties (DDS2) and (DDS1) imply that f(g(x)) = ϕ(1, ϕ(−1, x)) = ϕ(0, x) = x, x ∈ M. Thus, g(f(x)) = x, and g = f−1 . The equality ϕ(m, x) = f m (x) is an immediate corollary of property (DDS2). Thus, it is possible to define a dynamical system with discrete time taking as the initial object either a homeomorphism or a mapping with properties (DDS1)–(DDS3). https://doi.org/10.1515/9783110657166-001
- 22. 2 | 1 Dynamical systems These two approaches lead to the same result. For this reason, in what follows we do not distinguish a homeomorphism and the dynamical system generated by this homeomorphism. The basic object which is studied in theory of dynamical systems is defined as follows. Fix a homeomorphism f and a point x of the phase space. The trajectory of the point x in the dynamical system generated by f is the set O(x, f) = {fm (x) : m ∈ ℤ}. Sometimes, if the system is fixed, we denote a trajectory by O(x); if the point x is irrelevant, we use the notation O(f). Clearly, the following statement holds. Lemma 1.1. O(fm (x), f ) = O(x, f) for any m ∈ ℤ. We also apply the following notation: O+ (x, f) = {fm (x) : m ∈ ℤ+} and O− (x, f) = {fm (x) : m ∈ ℤ−}; the sets O+ (x, f) and O− (x, f) are called the positive and negative semitrajectories of the point x, respectively. Similar objects are defined for a subset A of the phase space; the set O(A, f) = {fm (A) : m ∈ ℤ} is called the trajectory of a set A in the dynamical system generated by a homeomor- phism f , and the sets O+ (A, f) = {fm (A) : m ∈ ℤ+} and O− (A, f ) = {fm (A) : m ∈ ℤ−} are called the positive and negative semitrajectories of the set A, respectively. It is easily shown that, for a trajectory of a discrete dynamical system, only one of the following three possibilities can be realized (this fact is a corollary of Lemma 1.2 below). 1. f (x) = x. In this case, the point x is called a fixed point; the trajectory of a fixed point coincides with the fixed point. 2. There exists a number m ∈ ℕ such that the points x, f(x), . . . , f m−1 (x) are distinct, and fm (x) = x. Such a point x is called periodic, the number m is called the period of the point x. The trajectory of x consists of m points x, f (x), . . . , fm−1 (x). Of course, a fixed point is periodic (with period 1); by tradition, fixed and periodic points (with period m > 1) are defined separately.
- 23. 1.1 Main definitions | 3 3. The points f l (x) and fm (x) are different if l ̸ = m. In this case, the trajectory of x is a countable set. Denote by Per(f) the set of periodic points of a homeomorphism f (we include fixed points in this set). Lemma 1.2. The set O(x, f) is finite if and only if x ∈ Per(f). Proof. It was mentioned above that if x ∈ Per(f ), then the set O(x, f ) is finite. Let us assume that the set O(x, f ) is finite. In this case, there exist different inte- ger numbers k and l such that f k (x) = f l (x). Let l > k; set n = l − k. Applying the homeomorphism f −k to the equality fk (x) = fl (x), we see that x = f n (x). If the points x, f(x), . . . , f n−1 (x) are distinct, then x is a periodic point of period n. Otherwise, either x is a fixed point of f or there exist different integer numbers k1, l1 ∈ [0, n − 1] such that f k1 (x) = fl1 (x). Let k1 < l1 and let n1 = l1 − k1. Then fn1 (y) = y, where y = fk1 (x) and n1 < n. Repeating this reasoning, we find a periodic point belonging to O(x, f); clearly, in this case, x is a periodic point of f. Now we introduce one more basic notion of theory of dynamical systems. We say that a set I ⊂ M is invariant for the dynamical system generated by a homeomorphism f if O(x, f) ⊂ I for any point x ∈ I. Lemma 1.3. A set I is invariant if and only if f(I) = I. Proof. Let I be an invariant set. Fix a point x ∈ I. Since O(x, f) ⊂ I, f (x) ∈ I and f−1 (x) ∈ I. Hence, f(I) ⊂ I and f −1 (I) ⊂ I (thus, I ⊂ f(I)); it follows that f (I) = I. Inverting the reasoning above, we see that if f(I) = I, then the set I is invariant. It follows from the well-known properties of homeomorphisms that if I and J are invariant sets, then the sets I ∪ J, I ∩ J, I J, Cl I, and 𝜕I are invariant as well. We give an important example of a dynamical system (we refer to this example below several times). Example 1.1. Let 𝒳 be the space whose elements are two-sided, infinite, binary se- quences a = {ai : ai ∈ {0, 1}, i ∈ ℤ}. We introduce the following metric in the space 𝒳: If a = {ai} and b = {bi}, we set dist(a, b) = ∞ ∑ i=−∞ |ai − bi| 2|i| (check that the above formula defines a metric). Clearly, our definition of the metric dist implies the following statement. For any given ϵ > 0 there exist numbers N(ϵ) and n(ϵ) such that if ai = bi, |i| ≤ N(ϵ),
- 24. 4 | 1 Dynamical systems then dist(a, b) < ϵ, and if dist(a, b) < ϵ, then ai = bi, |i| ≤ n(ϵ). Obviously, N(ϵ), n(ϵ) → ∞ as ϵ → 0. Let us recall the definition of the Tikhonov product topology (for the particular case of the space 𝒳) (see [12]). Fix an element a = {ai} of 𝒳 and a finite subset K ⊂ ℤ. Consider the cylinder C(a, K) = {b ∈ 𝒳 : bi = ai, i ∈ K}. Recall that a base of neighborhoods of a point x of a topological space is a family of neighborhoods of x such that any neighborhood of x contains a neighborhood from this family. The base of neighborhoods of a in the Tikhonov product topology consists of the cylinders C(a, K) corresponding to all finite subsets K ⊂ ℤ. It is easy to show that the metric topology induced by our metric dist coincides with the Tikhonov product topology (which means that the families of open sets in these topologies are the same). To prove this statement, it is enough to show that if b ∈ C(a, K) for some a ∈ 𝒳 and a finite set K ⊂ ℤ, then there is a small d > 0 such that the metric ball N(d, b) = {b ∈ 𝒳 : dist(b , b) < d} is a subset of C(a, K) and, conversely, if b ∈ N(d, a) for some d > 0, then there is a finite set K ⊂ ℤ such that C(b, K) ⊂ N(d, a) (we leave details to the reader). The metric space (𝒳, dist) is compact. This fact follows from the Tikhonov theo- rem [12] since the space 𝒳 is the countable product of compact spaces {0, 1} and, as was said, our metric dist induces on 𝒳 the Tikhonov product topology. Let us give an independent simple proof of the compactness of (𝒳, dist). It is known that a metric space is compact if and only if any sequence contains a conver- gent subsequence. Consider an arbitrary sequence am = {am i : i ∈ ℤ}, m ≥ 0. The elements am 0 take values 0 and 1; hence, there exists a subsequence m(0) = {m(0, 1), m(0, 2), . . . } of {0, 1, 2, . . . } such that 0 < m(0, 1) < m(0, 2) < ⋅ ⋅ ⋅ and am(0,1) 0 = am(0,2) 0 = ⋅ ⋅ ⋅ . Similarly, there exists a subsequence m(1) = {m(1, 1), m(1, 2), . . . }
- 25. 1.1 Main definitions | 5 of m(0) such that 0 < m(1, 1) < m(1, 2) < ⋅ ⋅ ⋅ and am(1,1) i = am(1,2) i = ⋅ ⋅ ⋅ , i = −1, 0, 1. Continuing this process, we find subsequences m(k) = {m(k, 1), m(k, 2), . . . } of the sequences m(k − 1) such that m(k, 1) < m(k, 2) < ⋅ ⋅ ⋅ and am(k,1) i = am(k,2) i = ⋅ ⋅ ⋅ , i = −k, . . . , k. Define elements bk , k ≥ 1, of the space 𝒳 by the equalities bk = am(k,1) . Clearly, the sequence {bk } is a subsequence of the sequence {am } with the following property: bk i = am(k,1) i = am(|i|,1) i , i = −k, . . . , k. Consider the element b of the space 𝒳 defined by the relations bi = am(|i|,1) i , i ∈ ℤ; then bi = bk i , i = −k, . . . , k, and we see that dist(bk , b) → 0, k → ∞. Thus, we have shown that the space (𝒳, dist) is compact. Consider a mapping σ of the space 𝒳 into itself defined as follows: We assign to an element a = {ai} of the space 𝒳 the element σ(a) = b = {bi} by the following rule: bi = ai+1, i ∈ ℤ. The mapping σ shifts indices by 1. Clearly, the mapping σ is invertible: σ−1 (a) = b if and only if bi = ai−1, i ∈ ℤ. Both mappings σ and σ−1 are continuous. Let us prove that σ is continuous. Take ϵ > 0 and find the corresponding number N(ϵ). Since n(δ) → ∞ as δ → 0, there exists a δ > 0 such that n(δ) > N(ϵ) + 1. If dist(a, b) < δ, then bi = ai, |i| ≤ n(δ). In this case, bi+1 = ai+1, |i| ≤ N(ϵ), and we conclude that dist(σ(a), σ(b)) < ϵ. A similar reasoning is applicable to σ−1 .
- 26. 6 | 1 Dynamical systems Thus, σ is a homeomorphism of the space 𝒳. The mapping σ (as well as the dy- namical system generated by this mapping) is usually called the shift on the space of binary sequences. Sometimes it is called the Bernoulli shift (though this last term may be applied to more complicated objects). Let us note several important properties of the shift. Property 1. The system σ has infinitely many different periodic points. Proof. Clearly, the equality σm (a) = b is equivalent to the relations bi = ai+m, i ∈ ℤ. Thus, σm (a) = a if and only if ai = ai+m, i ∈ ℤ. This means that the set of periodic points of the shift σ coincides with the set of peri- odic binary sequences, for which the statement is trivial. Property 2. The set of periodic points of σ is dense in the space 𝒳. Proof. Fix an arbitrary element a of the space 𝒳 and an arbitrary ϵ > 0. Find for this ϵ the corresponding number N(ϵ) and denote it by N. Let us construct a periodic binary sequence b as follows: represent any index i ∈ ℤ in the form i = k(2N + 1) + l, where k ∈ ℤ and |l| ≤ N, and set bi = al. Clearly, b is a periodic point of σ and bi = ai for |i| ≤ N, i. e., dist(a, b) < ϵ. Property 3. There exists an element of the space 𝒳 whose positive semitrajectory in the system σ is dense in the space 𝒳. Proof. Let us construct the desired element a as follows. Take arbitrary ai, i < 0. Set a0 = 0 and a1 = 1. Fix the pairs (a2, a3) = (0, 0), (a4, a5) = (0, 1), (a6, a7) = (1, 0), (a8, a9) = (1, 1); thus, a1 is followed from the right by all possible blocks of zeros and units of length 2. After that, we put to the right of a9 all possible blocks of zeros and units of length 3, and so on. Clearly, the element a has the following property: For any finite block (b1, . . . , bn) of zeros and units there exists an index k such that ak = b1, ak+1 = b2, . . . , ak+n−1 = bn. Let us prove that the closure of the semitrajectory O+ (a, σ) coincides with the space 𝒳. Fix arbitrary b ∈ 𝒳 and ϵ > 0. Find for the chosen ϵ the corresponding number N(ϵ) and denote it by N. By the construction of a, there exists an index k ≥ 0 such that ak = b−N , . . . , ak+2N = bN .
- 27. 1.1 Main definitions | 7 Set a = σ−k−N (a); then a i = bi for |i| ≤ N. This means that dist(a , b) < ϵ. Thus, we can find a point of O+ (a, σ) in an arbitrary neighborhood of an arbitrary element of 𝒳. We define the second basic class of dynamical systems axiomatically. Let, as above, M be a topological space. A mapping ϕ : ℝ × M → M is called a (continuous) dynamical system with contin- uous time (a flow) if this mapping has the following properties: (CDS1) ϕ(0, x) = x, x ∈ M; (CDS2) ϕ(t + s, x) = ϕ(t, ϕ(s, x)), t, s ∈ ℝ, x ∈ M; (CDS3) the mapping ϕ is continuous. In this case, the space M is called the phase space of the system. Sometimes, property (CDS3) is replaced by a weaker property: (CDS3 ) for any t ∈ ℝ, the mapping ϕ(t, ⋅) is continuous (such an assumption corresponds to the general notion of action of a group, which we consider in Section 1.5). In this book, we study flows with properties (CDS1)–(CDS3) (let us note that these properties are satisfied in the case of flows generated by autonomous systems of dif- ferential equations, the main class of flows which we study here). Along with continuous dynamical systems (with continuous and discrete time), we consider smooth dynamical systems, replacing the condition of continuity of the mapping ϕ in (DDS3) and (CDS3) by the condition of smoothness of this mapping (the exact smoothness conditions are stated separately in every particular case). Similarly to the case of a dynamical system with discrete time, we define the tra- jectory of a point x in the flow ϕ by the equality O(x, ϕ) = {ϕ(t, x) : t ∈ ℝ}. It is known (see the basic course of differential equations) that the following three types of trajectories of a flow are possible. Consider a point x0 ∈ M. 1. O(x0, ϕ) = {x0}. Such a trajectory (and the point x0 itself) is called a rest point. 2. O(x0, ϕ) ̸ = {x0}, and the mapping ϕ(t, x0) is periodic in t. In this case, the trajectory O(x0, ϕ) is called a closed trajectory of the flow ϕ. 3. ϕ(t, x0) ̸ = ϕ(s, x0) for s ̸ = t. In this case, the trajectory O(x0, ϕ) is a one-to-one image of the line. Similarly to the case of a discrete dynamical system, we say that a subset of the phase space is invariant under a flow if it contains trajectories of all its points. It is useful for the reader to formulate and prove an analog of Lemma 1.3 for flows. As was mentioned above, we mostly study flows generated by autonomous sys- tems of differential equations.
- 28. 8 | 1 Dynamical systems Consider an autonomous system of differential equations dx dt = F(x) (1) in the Euclidean space ℝn . We assume that the vector-function F is of class C1 in ℝn . Let x0 be an arbitrary point of the space ℝn . It is known from the basic course of differential equations that there exists a number h > 0 with the following property: On the interval (−h, h), there exists a unique solution ϕ(t, x0) of system (1) with initial data (0, x0). As usual, the graph of the mapping ϕ(⋅, x0) : (−h, h) → ℝn i. e., the set {(t, ϕ(t, x0)) : t ∈ (−h, h)} is called the integral curve of the solution ϕ(t, x0). The projection of the integral curve to the space ℝn , i. e., the set {x = ϕ(t, x0) : t ∈ (−h, h)} is called the trajectory of the solution ϕ(t, x0). Let us first assume that every maximally continued solution of system (1) is de- fined for t ∈ ℝ. In this case, the corresponding mapping ϕ : ℝ×ℝn → ℝn has properties (CDS1)–(CDS3); thus, this mapping is a flow. Property (CDS1) holds since ϕ(0, x) = x. Property (CDS2) is the group property of autonomous systems of differential equations (sometimes, this property is called the basic identity of autonomous systems). Under our assumptions on the smoothness of F, the mapping ϕ is continuous in (t, x) and differentiable in t and x (these statements are corollaries of the definition of a solution of a differential equation and of theorems on continuity and differentiability of a solution with respect to initial values). It is known that, in general, not every solution of a (nonlinear) system of differ- ential equations can be continued to the whole real line. To avoid this difficulty, the following idea can be used. Consider, along with system (1), the system dx dt = G(x), (2) where G(x) = F(x) 1 + |F(x)|2 and |x| is the Euclidean norm of a vector x (thus, if F = (F1, . . . , Fn), then |F(x)|2 = F2 1 + ⋅ ⋅ ⋅ + F2 n). Below we write F2 (x) instead of |F(x)|2 .
- 29. 1.1 Main definitions | 9 Clearly, the vector-function G is of class C1 , and the following inequality holds: G(x) < 1, x ∈ ℝn . (3) Let us denote by ψ(t, x) the trajectory of system (2) with initial condition ψ(0, x) = x. Inequality (3) implies that any maximally continued solution of system (2) is de- fined for t ∈ ℝ (why?). Thus, system (2) generates a flow in ℝn . Let us describe a relation between solutions of systems (1) and (2). Let y(t) be a solution of system (2) defined for t ∈ ℝ. Consider the function H(τ) = τ ∫ 0 ds 1 + F2(y(s)) defined for τ ∈ ℝ. Let V be the range of values of the function H. Since dH dτ > 0 for any t ∈ V, the equation t = H(τ) has a unique solution θ(t); clearly, the function θ is of class C1 . Differentiating the identity θ(H(τ)) ≡ τ in τ, we get the following identity: dθ dt dH dτ ≡ 1. Hence, dθ dt = ( 1 1 + F2(y(τ)) ) −1 = 1 + F2 (y(θ(t))). Let us check that the function z(t) = y(θ(t)) is a solution of system (1). Indeed, dz dt = dy dθ dθ dt = F(y(θ(t))) 1 + F2(y(θ(t))) (1 + F2 (y(θ(t)))) = F(z(t)). Thus, solutions of systems (1) and (2) (as well as trajectories of these solutions) differ by parametrization only; the structure of partition of the phase space into trajectories is the same for both systems. Since the structure of partition of the phase space into trajectories is the main ob- ject which we study in this book, in what follows we assume that autonomous systems of differential equations we work with generate flows. When one studies the global structure of dynamical systems, it is natural to con- sider systems on manifolds. We treat in detail smooth dynamical systems with dis- crete time generated by diffeomorphisms of smooth manifolds. For flows generated by
- 30. 10 | 1 Dynamical systems smooth vector fields, we formulate analogs of results established for smooth dynam- ical systems with discrete time. For this reason, we do not give here exact definitions from the theory of smooth vector fields on manifolds (the reader can find them, for example, in the book [13]). Let us briefly recall that a smooth tangent vector field F on a smooth manifold M is a smooth mapping of the manifold M into its tangent bundle TM. A smooth curve γ = ϕ(⋅, x) : I → M, where I is an interval of the real line, is called the trajectory of a point x ∈ M for the field F if ϕ(0, x) = x and the tangent vector of γ at the point ϕ(t, x) coincides with the vector F(ϕ(t, x)) for any t ∈ I. If the manifold M is compact, then any trajectory of a field F can be continued to ℝ; thus, in this case any vector field F generates a flow on M. Let us indicate several relations between the objects defined above and theories of vector fields and differential equations. 1.2 Embedding of a discrete dynamical system into a flow Let ϕ be a flow on a topological space M. Fix T > 0 and consider the mapping f : M → M defined by the formula f (x) = ϕ(T, x). Let us show that f is a homeomorphism of M. Indeed, if g(x) = ϕ(−T, x), then g is continuous (see property (CDS3)), and g(f(x)) = ϕ(−T, ϕ(T, x)) = ϕ(0, x) = x by properties (CDS2) and (CDS1). A similar reasoning shows that f(g(x)) = x. Hence, g is the inverse of f, and f is a homeomorphism. In this case, we say that the homeomorphism f is embedded into the flow ϕ. If M is a smooth manifold and the flow ϕ is smooth (which means that the map- pings ϕ(t, ⋅) are smooth for any t), then the same reasoning as above shows that any homeomorphism f embedded into the flow ϕ is a diffeomorphism. Clearly, the structure of the set of trajectories of a flow and the structure of the set of trajectories of a homeomorphism embedded into this flow are closely related. Nevertheless, one must remember that, in general, properties of the correspond- ing dynamical systems may differ significantly. Let us consider the following example. Example 1.2. Let S be the circle of unit length; introduce on S coordinate x ∈ [0, 1). Define a flow on S by the equality ϕ(t, x) = x + t (mod 1); in this flow, every point moves along the circle into positive direction with unit speed.
- 31. 1.3 Local Poincaré diffeomorphism | 11 Clearly, the flow ϕ has exactly one trajectory; this is a closed trajectory coinciding with the circle S. As was shown above, for any T > 0, the mapping f(x) = ϕ(T, x) is a homeomor- phism of the circle S. The dynamics of f is different for rational and irrational T. If T ∈ (0, 1) equals n/m, where n and m are relatively prime natural numbers, then any point x ∈ S is a periodic point of f of period m. If the number T is irrational, then the set of periodic points of f is empty, and every trajectory is a countable set of points that is dense in S (check this!) In addition, one has to remember that there exist diffeomorphisms that cannot be embedded into flows generated by smooth vector fields, and the set of such flows is large enough; this set is residual in the space of all diffeomorphisms (the exact defini- tions and statement of the result can be found in Section 2.4). 1.3 Local Poincaré diffeomorphism Consider system (1) and assume that a point p ∈ ℝn is not a rest point. Fix a number T > 0 and denote q = ϕ(T, p). Consider two smooth (n − 1)-dimensional surfaces P and Q in ℝn that contain the points p and q, respectively. We assume that locally (in neighborhoods of the points p and q) these surfaces are determined by smooth mappings Φ : ℝn−1 → ℝn and Ψ : ℝn−1 → ℝn , Φ, Ψ ∈ C1 , P is parametrized by parameter s ∈ ℝn−1 , Q is parametrized by parameter σ ∈ Rn−1 , and the equalities Φ(0) = p and Ψ(0) = q hold. We assume, in addition, that the surfaces P and Q are nondegenerate at the points p and q, respectively, which means that the ranks of the matrices A = 𝜕Φ 𝜕s (0) and B = 𝜕Ψ 𝜕σ (0) equal n − 1. Denote by a1, . . . , an−1 and b1, . . . , bn−1 the columns of the matrices A and B, respec- tively. In this case, the tangent spaces TpP of the surface P at the point p and TqQ of the surface Q at the point q are spanned by the vectors a1, . . . , an−1 and b1, . . . , bn−1, respectively. We say that the surfaces P and Q are transverse to the trajectory ϕ(t, p) at the points p and q if the tangent vectors F(p) and F(q) of the trajectory do not belong to the spaces TpP and TqQ, respectively.
- 32. 12 | 1 Dynamical systems Theorem 1.1. If the surfaces P and Q are transverse to the trajectory ϕ(t, p) at the points p and q, then the mapping determined by the shift along trajectories of system (1) is a diffeomorphism of a neighborhood of the point p in P to a neighborhood of the point q in Q. To prove Theorem 1.1, we apply a variant of the implicit function theorem, which we formulate below (Theorem 1.2). Consider two Euclidean spaces ℝl and ℝm with coordinates x and y, respectively. Theorem 1.2. Let f be a mapping of class C1 from a neighborhood of a point (a, b) in ℝl × ℝm to the space ℝm . Assume that f (a, b) = 0 and rank 𝜕f/𝜕y(a, b) = m. Then there exists a neighborhood U of the point a in ℝl and a mapping g of class C1 from the neighborhood U to ℝm such that g(a) = b and f(x, g(x)) = 0 for x ∈ U. Proof of Theorem 1.1. Since the vector F(p) does not belong to the space spanned by the vectors a1, . . . , an−1, rank(A, F(p)) = n. (4) Similarly, rank(B, F(q)) = n. (5) The trajectory of system (1) starting at a point Φ(s) ∈ P intersects the surface Q if and only if there exist t ∈ ℝ and σ ∈ ℝn−1 such that ϕ(t, Φ(s)) = Ψ(σ). Consider the function f(s, t, σ) = ϕ(t, Φ(s)) − Ψ(σ). This function maps a neighborhood of the point (0, T, 0) in ℝn−1 × ℝ × ℝn−1 to the space ℝn . Since the solution ϕ(t, x) is continuously differentiable in t and x, the func- tion f is of class C1 . In addition, f (0, T, 0) = ϕ(T, p) − q = 0. Let us calculate the Jacobi matrix 𝜕f 𝜕(t, σ) (0, T, 0) = ( 𝜕f 𝜕t , 𝜕f 𝜕σ )(0, T, 0) = ( 𝜕ϕ(t, Φ(s)) 𝜕t , − 𝜕Ψ(σ) 𝜕σ )(0, T, 0) = (F(q), −B). Equality (5) implies that f satisfies the conditions of Theorem 1.2 with l = n − 1, m = n, a = 0, and b = (T, 0). Hence, there exist mappings t(s) and σ(s) of class C1 defined for small |s| such that f(s, t(s), σ(s)) = 0, i. e., ϕ(t(s), Φ(s)) = Ψ(σ(s)),
- 33. 1.4 Time-periodic systems of differential equations | 13 t(0) = T, and σ(0) = 0. The mapping which assigns to points Φ(s) ∈ P with small |s| the points ϕ(t(s), Φ(s)) = Ψ(σ(s)) ∈ Q is differentiable and invertible (the existence and differentiability of the inverse mapping are proved similarly using equality (4)). The theorem is proven. Remark. We can apply the reasoning used in the proof of Theorem 1.1 to the function f (x, t, σ) = ϕ(t, x) − Ψ(σ), which maps a neighborhood of the point (p, T, 0) in ℝn × ℝ × ℝn−1 to the space ℝn and show that there exists a neighborhood U of the point p in ℝn and mappings t(x) and σ(x) of class C1 defined in U such that f (x, t(x), σ(x)) = 0, i. e., ϕ(t(x), x) = Ψ(σ(x)), and the following limit relations hold: t(x) → T and σ(x) → 0 as x → p. Thus, any trajectory that intersects a small neighborhood of the point p intersects the surface Q as well. The diffeomorphism given by Theorem 1.1 is called the local Poincaré diffeomor- phism generated by the transverse surfaces P and Q. The most important particular case of the construction described by Theorem 1.1 arises when the trajectory of the point p corresponds to a nonconstant periodic so- lution of system (1) (thus, it is a closed trajectory) and the surfaces P and Q coincide (precisely this case was studied by Poincaré). 1.4 Time-periodic systems of differential equations Consider a time-periodic system of differential equations, dx dt = F(t, x), (6) where x ∈ ℝn . We assume that the vector-function F is of class C0,1 t,x in ℝ × ℝn and F(t + ω, x) ≡ F(t, x) for some ω > 0. Denote by x(t, t0, x0) the solution of system (6) with initial values (t0, x0). It is well known that if x(t) is a solution of system (6) and k ∈ ℤ, then the function x(t + kω) is a solution as well. For definiteness, we assume that every solution of system (6) can be continued to ℝ. Consider the mapping T(ξ) = x(ω, 0, ξ). Let us show that T is a diffeomorphism of the space ℝn . Let U(ξ) = x(−ω, 0, ξ). Fix ξ ∈ ℝn and denote ξ = U(ξ).
- 34. 14 | 1 Dynamical systems Consider two solutions x1(t) = x(t, 0, ξ ) and x2(t) = x(t − ω, −ω, ξ ) of system (6) (the function x2(t) is a solution as the shift by −ω of the solution x(t, −ω, ξ )). Since x1(0) = ξ and x2(0) = x(−ω, −ω, ξ ) = ξ , the solutions x1(t) and x2(t) coin- cide. By uniqueness, x2(ω) = x(0, −ω, x(−ω, 0, ξ)) = ξ. Since x1(ω) = ξ, we get the equal- ity T(U(ξ)) = ξ, which shows that U is the inverse of T. The mappings T are U are differentiable; hence, T is a diffeomorphism (called the Poincaré diffeomorphism of system (6)). The following statement holds. Lemma 1.4. A solution x(t, 0, x0) of system (6) has period mω if and only if x0 is a fixed point of the diffeomorphism Tm . Proof. If a solution x(t, 0, x0) has period mω, then x(t, 0, x0) ≡ x(t + mω, 0, x0). Set t = 0 in the above identity to show that x0 = x(mω, 0, x0) = Tm (x0). Hence, x0 is a fixed point of Tm . Assume now that x0 is a fixed point of Tm , i. e., x0 = Tm (x0). Consider the solutions x1(t) = x(t, 0, x0) and x2(t) = x(t+mω, 0, x0). Since x1(0) = x0 and x2(0) = x(mω, 0, x0) = Tm (x0) = x0, the solutions coincide, which means that x(t, 0, x0) is mω-periodic. Thus, the important problem on the existence of periodic solutions of a system of differential equations is reduced to the problem on the existence of a fixed point of a diffeomorphism; modern mathematics has a wide class of methods for this problem. 1.5 Action of a group Let G be a group, i. e., a set with binary operation ∗ : G × G → G that satisfies the following axioms: (G1) the operation ∗ is associative, i. e., (a ∗ b) ∗ c = a ∗ (b ∗ c) for a, b, c ∈ G; (G2) there exists an identity element, i. e., an element e ∈ G such that a ∗ e = e ∗ a for a ∈ G; (G3) there exist inverse elements, i. e., for any a ∈ G there exists an element b ∈ G such that a ∗ b = e. The (left) action of a group G on a topological space M is a mapping ϕ : G × M → M with the following properties: (A1) ϕ(e, x) = x for x ∈ M; (A2) ϕ(a ∗ b, x) = ϕ(a, ϕ(b, x)) for a, b ∈ G and x ∈ M. Usually, continuous actions are considered, i. e., it is assumed that (A3) the mapping ϕ(a, ⋅) is continuous for any a ∈ G.
- 35. 1.5 Action of a group | 15 The trajectory (orbit) of a point x ∈ M under the action of the group G is the set O(x, G) = {ϕ(a, x) : a ∈ G}. Clearly, dynamical systems with discrete and continuous time are actions of the groups ℤ and ℝ, respectively (where ∗ denotes addition).
- 36. 2 Topologies on spaces of dynamical systems 2.1 C0 -topology Let (M, dist) be a compact metric space. If f and g are two homeomorphisms of the space M, we set ρ0(f, g) = max x∈M max(dist(f (x), g(x)), dist(f−1 (x), g−1 (x))). (7) It is easy to show that ρ0 is a metric on the space of homeomorphisms of the space M. We denote by H(M) the space of homeomorphisms of the space M with the met- ric ρ0; the topology induced by the metric ρ0 is called the C0 -topology. Lemma 2.1. The metric space H(M) is complete. Proof. Consider a sequence of homeomorphisms fm that is fundamental with respect to ρ0. This means that for any ϵ > 0 we can find an index m0 such that ρ0(fl, fk) < ϵ for k, l > m0. Then max x∈M dist(fl(x), fk(x)) < ϵ and max x∈M dist(f−1 l (x), f−1 k (x)) < ϵ for k, l > m0. Thus, the sequences fm and f−1 m are fundamental with respect to the uniform metric r(f, g) = max x∈M dist(f (x), g(x)). Since the space of continuous mappings is complete with respect to the uniform metric r, there exist continuous mappings f and g of the space M such that r(fm, f) → 0 and r(f−1 m , g) → 0 as m → ∞. Fix a point x ∈ M. Passing to the limit as m → ∞ in the equality fm(f −1 m (x)) = x, we see that f (g(x)) = x. Similarly, g(f(x)) = x. Thus, f is a homeomorphism of the space M, and g = f−1 . Clearly, ρ0(fm, f) → 0 as m → ∞. The proof is complete. Remark. It is easy to show that if we replace the metric ρ0 by the uniform metric r, then the appearing space of homeomorphisms will not be complete. Indeed, let M be the segment [0, 1]. Fix an integer m > 1 and consider a continuous mapping fm : [0, 1] → [0, 1] defined as follows: fm(0) = 0, fm(1) = 1, fm(1/3) = 1/m, https://doi.org/10.1515/9783110657166-002
- 37. 2.2 C1 -topology | 17 fm(2/3) = 1−1/m, and fm is affine on any of the segments [0, 1/3], [1/3, 2/3], and [2/3, 1]. Clearly, fm is a homeomorphism of the segment [0, 1]. It is easily seen that the inequality fm(x) − fn(x) ≤ max( 1 m , 1 n ), x ∈ [0, 1], holds for any m, n > 1; thus, the sequence fm is fundamental with respect to the uni- form metric r. Hence, this sequence converges with respect to r, and the limit function f equals 0 on [0, 1/3] and 1 on [2/3, 1]. Hence, f is not a homeomorphism. Now let ϕ and ψ be two flows on a compact metric space (M, dist). It was shown in Section 1.2 that, for any t ̸ = 0, the mapping ϕ(t, ⋅) is a homeomorphism of the space M (for t = 0, the mapping ϕ(0, ⋅) = Id is a homeomorphism as well). Define ρ0(ϕ, ψ) = max t∈[−1,1] ρ0(ϕ(t, ⋅), ψ(t, ⋅)). (8) Clearly, ρ0 is a metric on the space of flows on M. We denote by ℱ0 (M) the space of flows on M with the metric ρ0; similarly to the case of the space of homeomorphisms, the topology induced by the metric ρ0 is called the C0 -topology. The same reasoning as in the proof of Lemma 2.1 shows that ℱ0 (M) is a complete metric space. 2.2 C1 -topology Let M be a smooth closed (i. e., compact and boundaryless) manifold. To introduce the C1 -topology on the space of diffeomorphisms of M, we assume that M is a submanifold of the Euclidean space ℝN (a different, equivalent, approach to the definition of the C1 -topology based on local coordinates is described in [9]). No generality is lost assuming that M is a submanifold of a Euclidean space since, by the classical Whitney theorem, any smooth closed manifold can be embedded into a Euclidean space of appropriate dimension. If M is a submanifold of ℝN , for any point x ∈ M we can identify the tangent space TxM of M at x with a linear subspace of ℝN . Consider the metric dist on M induced by the Euclidean metric of the space ℝN . For a vector v ∈ TxM we denote by |v| its norm as the norm in the space ℝN . Let f and g be two diffeomorphisms of the manifold M. Define the value ρ0(f, g) by the same formula (7) as for homeomorphisms of a compact metric space. Take a point x of the manifold M and a vector v from the tangent space TxM. We consider the vectors Df (x)v ∈ Tf(x)M and Dg(x)v ∈ Tg(x)M as vectors of the same Eu- clidean space ℝN . Hence, the following values are defined: |Df (x)v − Dg(x)v| and Df(x) − Dg(x) = sup v∈TxM,|v|=1 Df(x)v − Dg(x)v .
- 38. 18 | 2 Topologies on spaces of dynamical systems Similarly, one defines the value Df−1 (x) − Dg−1 (x) = sup v∈TxM,|v|=1 Df−1 (x)v − Dg−1 (x)v . Introduce the number ρ1(f , g) = ρ0(f, g) + sup x∈M Df(x) − Dg(x) + sup x∈M Df −1 (x) − Dg−1 (x) . Clearly, ρ1 is a metric on the space of diffeomorphisms of the manifold M. We denote by Diff1 (M) the space of diffeomorphisms of M with metric ρ1; the topology induced by the metric ρ1 is called the C1 -topology. The standard reasoning (left to the reader) shows that (Diff1 (M), ρ1) is a complete metric space. Now we consider the space of smooth flows on M. We say that a flow ϕ : ℝ×M → M is smooth if for any t ∈ ℝ, the mapping ϕ(t, ⋅) is smooth (for our purposes, it is enough to assume that this mapping is of class C1 ; this assumption is satisfied if we consider a flow generated by a vector field of class C1 ). Our reasoning above (see Section 1.2) shows that if a flow ϕ : ℝ × M → M is smooth, then the mapping ϕ(t, ⋅) is a diffeomorphism of the manifold M for any t. If ϕ and ψ are two smooth flows on M, we set ρ1(ϕ, ψ) = max t∈[−1,1] ρ1(ϕ(t, ⋅), ψ(t, ⋅)). (9) It is easy to show that ρ1 is a metric on the space of smooth flows on M; we denote by ℱ1 (M) the space of smooth flows on M with the metric ρ1. Similarly to the case of diffeomorphisms, the topology induced by the metric ρ1 is called the C1 -topology. 2.3 Metrics on the space of systems of differential equations Considering flows generated by vector fields on smooth closed manifolds, we have introduced two metrics, ρ0 and ρ1. Defining these metrics, we estimated differences between trajectories of the flows with the same initial values and between the corre- sponding “variational flows” on time intervals of fixed length. Considering flows generated by autonomous systems of differential equations, it is natural to study metrics that are based on differences between right-hand sides of the systems rather than on differences between trajectories. As was mentioned in Section 1.1, we denote by |x| the Euclidean norm of a vector x ∈ ℝn . For an n × n matrix A, let us denote by ‖A‖ its operator norm generated by | ⋅ |, i. e., the value ‖A‖ = max |x|=1 |Ax|.
- 39. 2.3 Metrics on the space of systems of differential equations | 19 Let us give a rough estimate of the operator norm of a matrix A, which we use below. Assume that the entries of A are aij, i, j ∈ {1, . . . , n}, and |aij| ≤ M, i, j ∈ {1, . . . , n}. Let us write vectors x, y ∈ ℝn as x = (x1, . . . , xn) and y = (y1, . . . , yn), respectively. If y = Ax and |x| = 1, then y2 = n ∑ i=1 (ai1x1 + ⋅ ⋅ ⋅ + ainxn)2 ≤ (by the Cauchy inequality) ≤ n ∑ i=1 (a2 i1 + ⋅ ⋅ ⋅ + a2 in)(x2 1 + ⋅ ⋅ ⋅ + x2 n) = n ∑ i,j=1 a2 ij ≤ n2 M2 , and we conclude that ‖A‖ ≤ nM. (10) Consider two systems of differential equations, dx dt = F(x) (11) and dx dt = G(x) (12) in ℝn . We assume that the vector-functions F and G are of class C1 in ℝn . Denote by ϕ and ψ the flows generated by systems (11) and (12), respectively (as was mentioned above, we assume that every system of differential equations which we consider generates a flow). In addition, we assume that the Jacobi matrix 𝜕F/𝜕x of the vector-function F is bounded (in particular, this implies that F is globally Lipschitz continuous in ℝn ; de- note by L its global Lipschitz constant) and uniformly continuous in ℝn . The above assumptions on the Jacobi matrix 𝜕F/𝜕x do not look very natural; this is what we “pay” for working in the noncompact space ℝn (for a vector field of class C1 on a compact manifold, the corresponding assumptions are satisfied automatically). Finally, we assume that the values r0(F, G) = sup x∈ℝn F(x) − G(x)
- 40. 20 | 2 Topologies on spaces of dynamical systems and r1(F, G) = r0(F, G) + sup x∈ℝn 𝜕F 𝜕x (x) − 𝜕G 𝜕x (x) are finite (in fact, we study the case where these values tend to 0). Below, we refer to the following elementary estimate of the difference between solutions of two systems of differential equations (to prove this estimate, it is enough to represent the considered solutions of Cauchy problems in the form of solutions of equivalent integral equations and to apply the Gronwall lemma; a similar but more complicated estimate is applied in the following proof of Lemma 2.2). Consider two systems of differential equations, dx dt = f (t, x) (13) and dx dt = g(t, x), (14) where x ∈ ℝn . Assume that the vector-functions f and g are continuous in ℝn+1 , f is globally Lip- schitz continuous in x with constant L, and the value m = sup (t,x)∈ℝn+1 f (t, x) − g(t, x) is finite. If x(t) and y(t) are solutions of systems (13) and (14), respectively, that are defined on the same segment [a, b] and have the same initial values (t0, x0), where t0 ∈ [a, b], then x(t) − y(t) ≤ m exp(L(b − a)), t ∈ [a, b]. (15) Lemma 2.2. (1) If r0(F, G) → 0, then ρ0(ϕ, ψ) → 0. (2) If r1(F, G) → 0, then ρ1(ϕ, ψ) → 0. Proof. Let x be an arbitrary point of the space ℝn . Since ϕ(t, x) and ψ(t, x) are solutions of systems (11) and (12), respectively, with the same initial values (0, x), estimate (15) implies that ρ0(ϕ, ψ) = sup x∈ℝn max t∈[−1,1] ϕ(t, x) − ψ(t, x) ≤ r0(F, G) exp(L). This proves statement (1).
- 41. 2.3 Metrics on the space of systems of differential equations | 21 To estimate the value ρ1(ϕ, ψ), we again fix a point x ∈ ℝn and consider the deriva- tives of the flows ϕ and ψ with respect to initial values, Y(t) = 𝜕ϕ 𝜕x (t) and Z(t) = 𝜕ψ 𝜕x (t). Recall that these derivatives are matrix-valued solutions of the variational systems dY dt = Φ(t, Y), where Φ(t, Y) = 𝜕F 𝜕x (t, ϕ(t))Y, (16) and dZ dt = Ψ(t, Z), where Ψ(t, Z) = 𝜕G 𝜕x (t, ψ(t))Z, (17) respectively, with the same initial values Y(0) = Z(0) = E, where E is the identity matrix of size n × n. It was assumed that the Jacobi matrix 𝜕F/𝜕x is bounded; in addition, since r1(F, G) → 0, we may assume, for example, that r1(F, G) ≤ 1. Hence, there exists a number N > 0 such that 𝜕F 𝜕x ≤ N, 𝜕G 𝜕x ≤ N, x ∈ ℝn . (18) First we estimate the value ‖Z(t)‖ for t ∈ [−1, 1] (we are not going to get an exact estimate of the value ‖Z(t)‖; it is important for us to estimate this value by a constant depending on N and independent from the initial point x of the trajectory ψ(t, x)). If z is a column of the matrix Z, then dz dt = 𝜕G 𝜕x (t, ψ(t))z (19) and |z(0)| = 1. We take the scalar product of equality (19) and the vector z(t): ⟨ dz dt , z⟩ = ⟨ 𝜕G 𝜕x (t, ψ(t)), z⟩, (20) where ⟨⋅⟩ denotes scalar product. As above, we denote by z2 (t) the square of the Euclidean norm of z(t). Relations (18) and (20) imply that 1 2 d dt z2 = ⟨ dz dt , z⟩ = ⟨ 𝜕G 𝜕x (t, ψ(t))z, z⟩ ≤ 𝜕G 𝜕x (t, ψ(t))z |z| ≤ 𝜕G 𝜕x (t, ψ(t)) |z||z| ≤ Nz2 .
- 42. 22 | 2 Topologies on spaces of dynamical systems Since z(t) ̸ = 0, it follows from the last inequality that d dt (log z2 ) ≤ 2N. Integrating this inequality and taking into account that z2 (0) = 1, we get the estimate z2 (t) ≤ N2 1 := exp(2N), |t| ≤ 1. Thus, z(t) ≤ N1, and estimate (10) implies that Z(t) ≤ N2 := nN1, |t| ≤ 1. Fix an arbitrary ϵ > 0 and find a δ > 0 such that if |x − x | < δ, then 𝜕F 𝜕x (x) − 𝜕F 𝜕x (x ) ≤ ϵ. Clearly, if r1(F, G) → 0, then r0(F, G) → 0. Find a δ1 > 0 such that if r1(F, G) < δ1, then r0(F, G) exp(L) < δ. The same reasoning as in the proof of statement (1) shows that ϕ(t, x) − ψ(t, x) ≤ δ, |t| ≤ 1, for any x ∈ ℝn . Then 𝜕F 𝜕x (ϕ(t, x)) − 𝜕F 𝜕x (ψ(t, x)) ≤ ϵ, |t| ≤ 1, (21) for any x ∈ ℝn . Let us write down equivalent integral equations for Y and Z: Y(t) = E + t ∫ 0 Φ(s, Y(s)) ds (22) and Z(t) = E + t ∫ 0 Ψ(s, Z(s)) ds. (23)
- 43. 2.4 Generic properties | 23 Relations (22) and (23) imply that Y(t) − Z(t) ≤ t ∫ 0 Φ(s, Y(s)) − Ψ(s, Z(s)) ds . The integrand is estimated as follows: Φ(s, Y(s)) − Ψ(s, Z(s)) ≤ Φ(s, Y(s)) − Φ(s, Z(s)) + Φ(s, Z(s)) − Ψ(s, Z(s)) . It follows from estimates (18) that Φ(s, Y(s)) − Φ(s, Z(s)) = 𝜕F 𝜕x (ϕ(s, x))(Y(s) − Z(s)) ≤ N Y(s) − Z(s) . (24) Further, if r1(F, G) < δ1, then Φ(s, Z(s)) − Ψ(s, Z(s)) ≤ ( 𝜕F 𝜕x (ϕ(s, x)) − 𝜕F 𝜕x (ψ(s, x)))Z(s) + ( 𝜕F 𝜕x (ψ(s, x)) − 𝜕G 𝜕x (ψ(s, x)))Z(s) ≤ (ϵ + r1(F, G))N2. Combining these inequalities with estimate (24), we get the inequality Y(t) − Z(t) ≤ t ∫ 0 (N Y(s) − Z(s) + (ϵ + r1(F, G))N2) ds . Applying the Gronwall lemma to the above inequality, we conclude that if r1(F, G) < δ1, then Y(t) − Z(t) ≤ (ϵ + r1(F, G))N2 exp(N) for t ∈ [−1, 1]. Since ϵ is arbitrary, the last inequality implies statement (2) of our lemma. 2.4 Generic properties In the global theory of dynamical systems, it is important to study generic properties. Let X be a topological space. We say that a set A ⊂ X is residual if there exists a countable family of open subsets {An : n ∈ ℤ} of the space X such that ⋂ n∈ℤ An = A. (25) A property of elements of the space X is called generic if there exists a residual subset of X such that every element of this subset has this property. If a property is generic, we say that a generic element of the space X has this property. The following classical theorem was proven by Baire.
- 44. 24 | 2 Topologies on spaces of dynamical systems Theorem 2.1. If X is a complete metric space, then any residual subset is dense in X. A residual set is “large” in the topological sense. At the same time, a residual set can be “small” in the sense of measure. Let us show that there exists a residual subset of the real line ℝ whose Lebesgue measure is zero. Fix a countable dense subset {an, n = 0, 1, . . . } of the line (for example, consider the set of rational numbers). Take a natural number m and consider the set Am = ⋃ n≥0 (an − 1 m2n , an + 1 m2n ). The set Am is an open and dense subset of the line, and its Lebesgue measure can be estimated as follows: mes Am ≤ ∞ ∑ n=0 2 m2n = 4 m . The set A = ∩m>0Am is residual, and mes A = 0. It was mentioned in Section 1.2 that there exist diffeomorphisms that cannot be embedded into flows. The precise statement of the main result of [14] is as follows. Theorem 2.2. Let M be a smooth closed manifold whose dimension is not less than 2. Then a generic diffeomorphism in Diff1 (M) cannot be embedded into a flow generated by a Lipschitz continuous vector field on M. 2.5 Immersions and embeddings In this book, we consider two main classes of mappings studied in topology, immer- sions and embeddings. We mostly restrict ourselves to immersions and embeddings of smooth manifolds (Euclidean spaces and disks in such spaces) into Euclidean spaces. Let us recall the basic definitions. Let X and Y be topological spaces. A mapping f from X to Y is a topological im- mersion if any point x ∈ X has a neighborhood U such that f maps homeomorphically U onto f (U). Now let f be a mapping of a smooth manifold M to a smooth manifold N. We say that f is an immersion of class Ck , k ≥ 1, if f belongs to class Ck and rank Df (x) = dim M for any x ∈ M (let us note that the last condition implies that dim N ≥ dim M). The mapping f is called an embedding of class Ck , k ≥ 1, if f (M) is a submanifold of N and f is a diffeomorphism of M and f(M). We will work with smooth disks. By definition, a smooth disk is the image of a ball of a Euclidean space under an embedding.
- 45. 2.5 Immersions and embeddings | 25 Consider a disk D = {x ∈ ℝk : |x| < r, r > 0} in the Euclidean space ℝk and a manifold M (as above, we assume that M is a sub- manifold of a Euclidean space ℝN ). For two embeddings h and g of the disk D into the manifold M we set ρ1(h, g) = sup x∈D ( h(x) − g(x) + Dh(x) − Dg(x) ), where | ⋅ | and ‖ ⋅ ‖ are the distance in ℝN and the corresponding operator norm (to show that all the objects are properly defined, one can use the same reasoning as in Section 2.2).
- 46. 3 Equivalence relations 3.1 Topological conjugacy Consider two homeomorphisms f : M → M and g : N → N, where M and N are topological spaces. We say that the homeomorphisms f are g topologically conjugate if there exists a homeomorphism h of the spaces M and N such that g(h(x)) = h(f(x)) (26) for any x ∈ M (in other words, g ∘ h = h ∘ f). In this case, the homeomorphism h is called a conjugating homeomorphism (or topological conjugacy). Sometimes, condition (26) is formulated in the following (equivalent) form: The diagram M f → M ↓ h ↓ h N g → N commutes. The following simple (but very important) statement holds. Lemma 3.1. If g ∘ h = h ∘ f, then gm ∘ h = h ∘ f m (27) for any m ∈ ℤ. Proof. We apply induction to prove the statement for m ≥ 0 (for m < 0, the proof is similar). If m = 0, f 0 = g0 = Id, and equality (27) takes the form h = h. Assume that equality (27) has been proven for some m. Then gm+1 ∘ h = g ∘ (gm ∘ h) = g ∘ (h ∘ fm ) = (g ∘ h) ∘ fm = (h ∘ f ) ∘ fm = h ∘ fm+1 , which proves the statement of our lemma. Lemma 3.1 implies that a conjugating homeomorphism maps trajectories of the dynamical system generated by the homeomorphism f to trajectories of the dynamical system generated by the homeomorphism g. Thus, if homeomorphisms f and g are topologically conjugate, then, from the topological point of view, the global structure of the set of trajectories of the dynamical systems generated by the homeomorphisms f and g is the same. https://doi.org/10.1515/9783110657166-003
- 47. 3.1 Topological conjugacy | 27 For example, periodic trajectories of the homeomorphism f are mapped to peri- odic trajectories of the homeomorphism g. Indeed, let p be a periodic point of f of period m, i. e., the points p0 = p, p1 = f(p), . . . , pm−1 = fm−1 (p) are distinct and fm (p) = p. If r = h(p), then ri = gi (r) = gi (h(p)) = h(f i (p)) = h(pi) by Lemma 3.1; thus, the points ri, i = 0, . . . , m − 1, are distinct and gm (r) = h(f m (p)) = h(p) = r. A similar reasoning shows that if a trajectory O(p, f ) (a semitrajectory O+ (p, f)) is dense in M, then the trajectory O(h(p), g) (semitrajectory O+ (h(p), g)) is dense in N. Remark. Sometimes, it is possible to significantly simplify a problem by passing from a homeomorphism to a topologically conjugate homeomorphism (this idea will be ap- plied in Chapter 9, where we study the Smale horseshoe). Here we consider as an example two semidynamical systems. Let f be a continuous mapping of a topological space M into itself. We set ϕ(m, x) = fm (x), m ∈ ℤ+, and get a semidynamical system, i. e., a mapping ϕ : ℤ+ × M → M whose properties are similar to properties (DDS1)–(DDS3) (one has to replace ℤ by ℤ+ in properties (DDS2) and (DDS3)). The trajectory of a point x in the semidynamical system ϕ is defined by the equality O(x, ϕ) = {ϕ(m, x) : m ∈ ℤ+}; the definition of a periodic point is literally the same as in the case of a dynamical system. Two semidynamical systems generated by mappings f and g are called topologi- cally conjugate if there exists a homeomorphism h that satisfies equality (26). Exercise 3.1. Consider two semidynamical systems on the segment [0, 1] generated by the mappings f(x) = 4x(1 − x) and g(x) = { 2x, x ∈ [0, 1/2], 2(1 − x), x ∈ (1/2, 1]. Prove that the mapping h(x) = 2 π arcsin √x
- 48. 28 | 3 Equivalence relations is a homeomorphism of the segment [0, 1] that conjugates the semidynamical systems generated by the mappings f and g. Thus, we can reduce the study of the dynamics of the essentially nonlinear map- ping f to the similar problem for the piecewise linear mapping g. In what follows, we consider dynamical systems with the same phase space M. Lemma 3.2. Topological conjugacy is an equivalence relation on the space H(M). Proof. Since the identity homeomorphism Id conjugates any homeomorphism with itself, topological conjugacy is reflexive. Topological conjugacy is symmetric. Indeed, if a homeomorphism h conjugates f and g, i. e., g ∘ h = h ∘ f, then h−1 conjugates g and f ; to show this, apply h−1 to the equality g∘h = h∘f both from the right and left. As a result, we get the desired equality f ∘ h−1 = h−1 ∘ g. Finally, we show that topological conjugacy is transitive. Assume that h1 conju- gates f and g and h2 conjugates g and k. Then h = h2 ∘ h1 is a homeomorphism of the space M, and h ∘ f = h2 ∘ (h1 ∘ f) = h2 ∘ (g ∘ h1) = (k ∘ h2) ∘ h1 = k ∘ h, which completes the proof of our lemma. Of course, topological conjugacy is an equivalence relation on the space of diffeo- morphisms Diff1 (M) of a smooth manifold M as well. This relation allows us to give the main definition of the theory of structural sta- bility. Let M be a smooth closed manifold. A diffeomorphism f ∈ Diff1 (M) is called structurally stable if there exists a neighborhood W of the diffeomorphism f in the C1 -topology such that any diffeomorphism g ∈ W is topologically conjugate with f. The above definition and Lemma 3.2 imply that any diffeomorphism g ∈ W is structurally stable as well. Denote by 𝒮(M) the set of structurally stable diffeomor- phisms in Diff1 (M). Clearly, the following statement holds (since this statement is very important for us, we call it a theorem). Theorem 3.1. The set 𝒮(M) is open in Diff1 (M). The property of structural stability was first defined by Andronov and Pontryagin for autonomous systems of differential equations (we discuss this definition below, in Section 7.6). In fact, this original definition corresponds to a slightly different property which we formulate below. A diffeomorphism f ∈ Diff1 (M) is called structurally stable in the strong sense if for any ϵ > 0 one can find a neighborhood W of the diffeomorphism f in the C1 -topology
- 49. 3.1 Topological conjugacy | 29 such that for any diffeomorphism g ∈ W there exists a homeomorphism h that topo- logically conjugates g and f and satisfies the inequality max x∈M dist(h(x), x) < ϵ. It is easy to understand that this definition does not imply immediately that the set of diffeomorphisms that are structurally stable in the strong sense is open. Let us pass to the case of flows. Consider two flows ϕ : ℝ×M → M and ψ : ℝ×N → N, where M and N are topological spaces. The flows ϕ and ψ are called topologically conjugate if there exists a homeomor- phism h of the spaces M and N such that ψ(t, h(x)) = h(ϕ(t, x)) (28) for any t ∈ ℝ and x ∈ M. Thus, topological conjugacy of flows means that there exists a homeomorphism of their phase spaces that maps trajectories to trajectories and preserves time t. Let us show that the notion of topological conjugacy of flows is too fine for the problem of global classification of flows generated by systems of differential equa- tions. Consider, for example, two autonomous systems of differential equations in the plane ℝ2 with coordinates (x, y): dx dt = −2y, dy dt = 2x, (29) and dx dt = −y, dy dt = x. (30) Let ϕ and ψ be the flows generated by systems (29) and (30), respectively. The origin is a rest point of both systems (29) and (30); the remaining trajectories are concentric circles with center at the origin along which points move in the positive direction as t grows. Fix an initial point (x0, 0). The trajectories of this point in the flows ϕ and ψ are given by the following formulas: ϕ(t, x0, 0) : x = x0 cos 2t, y = x0 sin 2t, and ψ(t, x0, 0) : x = x0 cos t, y = x0 sin t, respectively.
- 50. 30 | 3 Equivalence relations Clearly, the sets of trajectories of systems (29) and (30) are the same from the topo- logical point of view; at the same time, the flows ϕ and ψ are not topologically conju- gate. Let us show this. To get a contradiction, let us assume that there exists a homeomorphism h : ℝ2 → ℝ2 for which equality (28) holds. The trajectory of the point ϕ(t, 1, 0) is closed; hence, the homeomorphism h must map this trajectory to a closed trajectory. Thus, if h(1, 0) = (x0, y0), then (x0, y0) ̸ = (0, 0). (31) The formula defining the flow ϕ implies that ϕ(π, 1, 0) = (1, 0). Hence, (x0, y0) = h(1, 0) = h(ϕ(π, 1, 0)) = ψ(π, h(1, 0)) = ψ(π, x0, y0) = −(x0, y0), and we get a contradiction with inequality (31). In the problem of global classification of flows, a different notion of equivalence is used. We discuss this property in the next section. 3.2 Topological equivalence of flows Two flows ϕ : ℝ × M → M and ψ : ℝ × N → N, where M and N are topological spaces, are called topologically equivalent if there exists a homeomorphism h of the spaces M and N that maps trajectories of the flow ϕ to trajectories of the flow ψ and preserves the direction of movement along trajectories. In other words, there exists a function τ : ℝ × M → ℝ such that (1) for any x ∈ M, the function τ(⋅, x) increases and maps ℝ onto ℝ; (2) τ(0, x) = 0 for any x ∈ M; (3) h(ϕ(t, x)) = ψ(τ(t, x), h(x)) for any (t, x) ∈ ℝ × M. Clearly, the flows ϕ and ψ generated by systems of differential equations (29) and (30) are topologically equivalent; one may take as h the identical mapping of the plane and set τ(t, x) = 2t. 3.3 Nonwandering set Some equivalence relations which are important for the global qualitative theory of dynamical systems are related to the notion of a nonwandering point. Consider a homeomorphism f of a topological space M and the corresponding dynamical system. A point x0 ∈ M is called wandering for f if there exists a neighborhood U of the point x0 and a number N > 0 such that fn (U) ∩ U = 0 for |n| ≥ N.
- 51. 3.3 Nonwandering set | 31 A point x0 is called nonwandering if it is not wandering. Clearly, a point x0 is non- wandering if for any neighborhood U of x0 and for any number N there exist a point x ∈ U and a number n, |n| > N, such that fn (x) ∈ U. We denote by Ω(f ) the set of nonwandering points of a homeomorphism f (usually, the set Ω(f) is called the nonwandering set). Under rather general assumptions on the space M (for example, these assump- tions are satisfied for a metric space), we can give a different definition of a nonwan- dering point. A topological space M is said to satisfy the first axiom of countability if any point of M has a countable base of neighborhoods (we gave a definition of a base of neigh- borhoods in Section 1.1 when Example 1.1 was considered). It is well known that any metric space satisfies the first axiom of countability. Lemma 3.3. Assume that the space M satisfies the first axiom of countability. A point x0 ∈ M is nonwandering for a homeomorphism f if and only if there exist sequences of points pk ∈ M and numbers τk such that pk, f τk (pk) → x0 and τk → ∞ as k → ∞. Proof. Clearly, if such sequences exist, then x0 ∈ Ω(f). Take a point x0 ∈ Ω(f) and fix a countable base Vm, m > 0, of neighborhoods of the point x0 such that x0 = ∩m>0Vm. For any natural m we can find a number n(m) such that |n(m)| > m and fn(m) (Vm) ∩ Vm ̸ = 0. This means that there exist points rm ∈ Vm such that f n(m) (rm) ∈ Vm; thus, rm, f n(m) (rm) → x0, m → ∞. If the sequence n(m) contains a subsequence n(mk) → ∞, we set pk = rn(mk) and τk = n(mk); otherwise (if n(m) → −∞), we set pk = rk and τk = −n(k). Clearly, fixed and periodic points of a homeomorphism f are nonwandering. In- deed, if p is a periodic point of period m, then the points f mk (p) = p belong to any neighborhood of p, while the numbers mk can be arbitrarily large. Sometimes, such nonwandering points are called trivial. There exist nontrivial nonwandering points. Let us recall the notions of ω-limit and α-limit sets of a trajectory O(x, f). The ω-limit set, ω(x, f), of a trajectory O(x, f) is, by definition, the set of limit points of all sequences fn(k) (x), where n(k) → ∞ as k → ∞. Similarly, the α-limit set, α(x, f), of a trajectory O(x, f ) is, by definition, the set of limit points of all sequences f n(k) (x), where n(k) → −∞ as k → ∞. The two sets ω(x, f ) and α(x, f) are closed and invariant.
- 52. 32 | 3 Equivalence relations Lemma 3.4. ω(x, f ) ∪ α(x, f ) ⊂ Ω(f ) for any point x ∈ M. Proof. Let us prove that ω(x, f) ⊂ Ω(f ); the case of α-limit set is considered analo- gously. Take a point x0 ∈ ω(x, f ). There exists a sequence n(k) → ∞, k → ∞, such that fn(k) (x) → x0. Let U be an arbitrary neighborhood of the point x0 and let N be an arbitrary num- ber. There exists an index k0 such that f n(k) (x) ∈ U for k ≥ k0. In addition, there exists an index k1 > k0 such that n1 := n(k1) − n(k0) > N. In this case, fn(k1) (x) = fn1 (f n(k0) (x)) ∈ U, i. e., fn1 (U) ∩ U ̸ = 0. This means that x0 ∈ Ω(f ). One can show that there exist dynamical systems for which the nonwandering set contains points that are not ω-limit or α-limit points of individual trajectories. Below we give an example of a flow having this property (see Example 3.1); let us mention that some notions and constructions are more “visible” in the case of a flow. Let us describe the basic properties of nonwandering sets. Theorem 3.2. The set Ω(f) is closed and invariant. If the space M is compact, then Ω(f) ̸ = 0. Proof. First we show that the set Ω(f ) is closed. It follows from the definition that if x0 is a wandering point, then any point of the neighborhood U mentioned in the definition is wandering as well. Thus, the set of wandering points is open; its complement Ω(f) is closed. Now we prove that the set Ω(f ) is invariant. Consider an arbitrary point x0 ∈ Ω(f ), an arbitrary neighborhood U of the point x = f(x0), and an arbitrary number N. Since the mapping f is continuous, the set U1 = f −1 (U) is a neighborhood of the point x0. Hence, there exist a point x1 ∈ U1 and a number n, |n| > N, such that f n (x1) ∈ U1. Let x = f(x1). Then x ∈ U and f n (x) = f(f n (x1)) ∈ f (U1) = U. Thus, x ∈ Ω(f ); it follows that f(Ω(f)) ⊂ Ω(f). A similar reasoning shows that f −1 (Ω(f)) ⊂ Ω(f ). Hence, f(Ω(f)) = Ω(f). Thus, the set Ω(f) is invariant by Lemma 1.3. Now let us assume that the space M is compact. In this case, the ω-limit set of any trajectory is nonempty, and the last statement of our theorem is a corollary of Lemma 3.4. It is easy to understand that if the phase space of a dynamical system is not com- pact, then the nonwandering set may be empty. As an example, consider the homeomorphism f(x) = x + 1 of the line ℝ. In a sense, the global dynamics is characterized by the behavior of a dynamical system near its nonwandering set. In fact, for any trajectory, only a finite number of its points does not belong to a neighborhood of the nonwandering set. More precisely, the following theorem was proven by Birkhoff (the constant T whose existence is es-
- 53. 3.3 Nonwandering set | 33 tablished in Theorem 3.3 is usually called the Birkhoff constant for a neighborhood U of the set Ω(f)). Theorem 3.3. Assume that the phase space M of a dynamical system generated by a homeomorphism f is compact. Let U be an arbitrary neighborhood of the set Ω(f). There exists a number T > 0 such that card{k : f k (x) ∉ U} ≤ T for any point x ∈ M. Proof. Fix a neighborhood U of the set Ω(f ). For any point x ∈ M U we can find a number t(x) and neighborhood V(x) such that f k (V(x)) ∩ V(x) = 0, |k| ≥ t(x). Since the set M U is compact, the covering {V(x)} of the set M U contains a finite subcovering V1, . . . , Vn with the following property: There exist numbers t1, . . . , tn ≥ 1 such that f k (Vi) ∩ Vi = 0, k ≥ ti, i = 1, . . . , n. Take a natural t such that t ≥ max ti and let T = nt. Let us prove that T has the required property. To get a contradiction, assume the contrary. Then there exist a point x and a set of integers L = {l(0), l(1), . . . , l(m)} with l(0) < l(1) < ⋅ ⋅ ⋅ < l(m) such that m + 1 > T and fl(i) (x) ∉ U for i = 0, . . . , m. Note that if i, j ∈ {0, . . . , m} and j ≥ i, then l(j) − l(i) ≥ j − i. The inequality card L = m + 1 > T = nt implies that the set L contains numbers l(jt) with j = 0, . . . , n. Consider points yj = fl(jt) (x) with j = 0, . . . , n and let Wj be neighborhoods from the family {V1, V2, . . . , Vn} that contain the points yj (if such a neighborhood is not unique, take as Wj any of them). If 0 ≤ j < k ≤ n, then l(kt) − l(jt) ≥ (k − j)t ≥ t; thus, if yj belongs to a neighborhood Vi from the family {V1, V2, . . . , Vn}, then yk = fl(kt)−l(jt) (yj) cannot belong to the same neighborhood Vi. It follows that if 0 ≤ j < k ≤ n, then Wj and Wk are different elements of the set {V1, V2, . . . , Vn}.
- 54. 34 | 3 Equivalence relations Thus, we have found n + 1 different elements in a set consisting of n elements, which is impossible. The contradiction obtained completes the proof. Consider two homeomorphisms f : M → M and g : N → N. We say that the homeomorphisms f and g are Ω-conjugate if there exists a homeomorphism h of Ω(f ) and Ω(g) such that g(h(x)) = h(f (x)) for x ∈ Ω(f) (let us explain that in this case h is a one-to-one mapping of Ω(f) onto Ω(g) such that both h and h−1 are continuous with respect to topologies induced on the sets Ω(f ) and Ω(g) by the topologies of the spaces M and N). Lemma 3.5. If h topologically conjugates f and g, then h(Ω(f)) = Ω(g). Proof. Consideranarbitrarypoint x0 ∈ Ω(f) andfixan arbitraryneighborhoodU ofthe point y0 = h(x0) and an arbitrary number N. Since h is continuous, the set V = h−1 (U) is a neighborhood of the point x0. Hence, there exist a point x ∈ V and a number n, |n| > N, such that f n (x) ∈ V. Denote y = h(x). Then y ∈ U and gn (y) = gn (h(x)) = h(fn (x)) ∈ U. Thus, y ∈ Ω(g); hence, h(Ω(f)) ⊂ Ω(g). A similar reasoning shows that h−1 (Ω(g)) ⊂ Ω(f ). We conclude that h(Ω(f)) = Ω(g). Corollary. If homeomorphisms f and g are topologically conjugate, then they are Ω-conjugate. The same reasoning as in the proof of Lemma 3.2 shows that Ω-conjugacy is an equivalence relation on the space of homeomorphisms H(M). Let M be a smooth closed manifold. A diffeomorphism f ∈ Diff1 (M) is called Ω-stable if there exists a neighborhood W of the diffeomorphism f in the C1 -topology such that any diffeomorphism g ∈ W is Ω-conjugate with f. The corollary of Lemma 3.5 implies that if a diffeomorphism is structurally stable, then it is Ω-stable. Nonwandering points of flows are defined similarly to nonwandering points of cascades. Let ϕ be a flow on a topological space M. A point x0 ∈ M is called a nonwandering point of the flow ϕ if for any neighborhood U of the point x0 and for any number N there exist a point x ∈ U and a number t, |t| > N, such that ϕ(t, x) ∈ U. The set of nonwandering points of a flow has the same basic properties as the set of nonwandering points of a cascade. Let us give an example of a flow whose nonwandering set does not coincide with the union of the sets of ω-limit and α-limit points of individual trajectories. Example 3.1. Consider the following autonomous system of differential equations in the plane ℝ2 with coordinates (x, y): dx dt = y, dy dt = x(1 − x2 ). (32)
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- 56. I open my thumbed copy of Modern Painters, turn to a certain page in volume I., and read this: '"The Snowstorm," one of the very grandest statements of sea-motion, mist, and light, that has ever been put on canvas, even by Turner.' In this appreciation we can go all the way with Ruskin. 'The Snowstorm' in its new home in the new Turner Gallery looks the work of a giant in the interpretation of sea-motion, mist and light. The 'Snowstorm; Steamboat off a Harbour's Mouth making Signals in shallow water and going by the lead,' was laughed at by the press when it was shown in the 1842 Academy. The parody of the title that appeared in Punch was almost funny; but the old man did not think it funny: 'A Typhoon bursting in a Simoon over the Whirlpool of Maelstrom, Norway; with a ship on fire, an eelipse, and the effect of a lunar rainbow,' with the following skit on the Fallacies of Hope:— 'O Art, how vast thy mighty wonders are To those who roam upon the extraordinary deep; Maelstrom, thy hand is here,'
- 57. Plate XXXIII. The Snow Storm (1842) Thornbury asserts that the critics of all kinds, learned and unlearned, were furious when it was exhibited; some of them described it as a mass of 'soapsuds and whitewash.' 'Turner,' wrote Ruskin, 'was passing the evening at my father's house, on the day this criticism came out; and after dinner, sitting in his arm-chair by the fire, I heard him muttering low to himself, at intervals, "Soapsuds and whitewash" again, and again, and again. At last I went to him, asking why he minded what they said. Then he burst out, "Soapsuds and whitewash! What would they have? I wonder what they think the sea's like? I wish they'd been in it."'
- 58. As a matter of fact, Turner had given himself infinitely more trouble over 'The Snowstorm' than over 'The Fighting Téméraire,' and he had been in considerable danger. To paint 'The Snowstorm,' he had put to sea from Harwich in the Ariel in a hurricane, had made the sailors lash him to the mast, and there the student of sixty-seven remained for four hours studying the awful scene. I look at 'The Snowstorm' to-day, and remember. I am filled with awe at the man's power. No, we do not smile at 'The Snowstorm' now; but certain folk still smile at 'War: The Exile and the Rock Limpet,' depicting an attenuated Napoleon, standing against a blood-red sunset, in the shallows of a tidal pool, on the shore of St. Helena, gazing with folded arms out to sea. Turner failed to make this nobly inspired dream a reality—that is all. Punch made merry over the 'Exile and the Rock Limpet,' calling it 'The Duke of Wellington and the Shrimp (Seringapatam, early morning),' with another parody of the Fallacies:— 'And can it be, thou hideous imp, That life is, ah! how brief, and glory but a shrimp! (From an unpublished poem.)' And remarked that:— 'The comet just rising above the cataract in the foreground, and the conflagration of Tippoo's widow in the Banyan forest by the sea-shore, are in the great artist's happiest manner.' 'Peace, Burial at Sea of the Body of Sir David Wilkie,' was a vision which Turner completely realised, the poetry, the pathos, the grandeur, the decorative splendour—all. The sails of the steamship are dark against the evening sky, as if in mourning, and amidships, in a blaze of torchlight, the body of Wilkie is being lowered to his watery grave. Stanfield, who saw the picture on Varnishing Day, thought the effect of the sails was 'untrue,' which, of course, they are, but Turner would not alter them. 'I only wish I had any colour to make them blacker,' said the old warrior. From this picture of peace and solemnity I turn to the peace and loveliness of some 'smaller' water-colours of this, his sunset, period.
- 59. PLATE XXXIV. Peace. Burial at Sea of Sir David Wilkie (1842) Tate Gallery Ruskin, in his 'Notes on Turner's Drawings exhibited at the Fine Art Society in 1878,' which is printed as the Epilogue to the volume called Notes on Pictures, tells how in the winter of 1841-42 Turner brought back with him from Switzerland a series of sketches, fifteen of which he placed, as was his custom, in the hands of his agent, Griffith of Norwood, so that he might obtain commissions for finished drawings of each.
- 60. Ruskin tells us that 'he made anticipatorily four, to manifest what their quality would be, and honestly "show his hand." Four thus exemplary drawings I say he made for specimens, or signs, as it were, for his re- opened shop, namely:— 1. The Pass of Splugen. 2. Mont Righi, seen from Lucerne, in the morning, dark against dawn. 3. Mont Righi, seen from Lucerne at evening, red with the last rays of sunset. 4. Lake Lucerne (The Bay of Uri) from above Brunnen, with exquisite blue and rose mists and 'mackerel' sky on the right. The whole story, which is told in Ruskin's most simple and charming style, is too long to be repeated here. Nine commissions only could be obtained, making ten with the one given to Griffith as commission. 'Turner growled, but said at last that he would do them,' and among them was a 'Lucerne Town,' which Ruskin, by hard coaxing and petitioning, obtained his father's leave to promise to take if it turned out well. It did. What a wonderful realisation of a dream of colour is another water- colour of this period, reproduced in these pages—'Spietz on the Lake of Thun, Looking Towards the Bernese Oberland.' On the last page of the Ruskin Catalogue, which is now called Epilogue, the old man, most eloquent and most sorrowful, writes:— 'The "Constance" and "Coblentz" here with the "Splugen" (1), "Bay of Uri" (4), and "Zurich" (10), of the year 1812, are the most finished and faultless works of his last period; but these of 1843 are the truest and mightiest ... I can't write any more of them just now.' About this time Munro of Novar offered twenty-five thousand pounds for the whole contents of the Queen Anne Street Gallery. Turner
- 61. hesitated, but finally refused. Frith, in his Autobiography, tells the story thus:— 'When Munro of Novar went for his final answer, Turner cried, "No! I won't—I can't. I believe I am going to die, and I intend to be buried in those two (pointing to "Carthage" and "The Sun Rising Through Vapour"), so I can't—besides I can't be bothered. Good-evening!"' The evening of his life was to last nine years, and Turner found his own way of escape from being bothered. CHAPTER XLIX 1843: AGED SIXTY-EIGHT VISIONS OF VENICE AND THE FIRST VOLUME OF 'MODERN PAINTERS' The two pictures of Venice exhibited in 1843, so changed, so faded, are in their way among the loveliest things Turner ever painted. 'San Benedetto, Looking Towards Fusina,' was formerly known as 'The Approach to Venice,' and I wish that title could have been retained, as one always thinks of it as 'The Approach to Venice,' and always in connection with the companion picture, 'The "Sun of Venice" Going to Sea,' with the name of this immortalised, fishing-boat 'Sol di Venezia' conspicuous on the sail. These two fading visions of Venice are indescribable, although everybody attempts to describe them. An eloquent passage may be found in the essay M. de la Sizeranne wrote for The Studio on 'The Genius of Turner,' from which the following is an extract:— 'Nothing will be found more beautiful than the "Approach" itself. No robe from Tintoretto's brush will be found to possess the
- 62. splendour of the gondolas conveying us. No Titian—that of the mountains of Cadore, the presence of which we divine, no nimbus about the head of a saint, will equal that sun, no purple these skies, no prayer the infinite sweetness of the dream experienced during those brief, delicious moments. Nothing will be found to compare with the distant vision of that city which, on the horizon, seems to be too beautiful ever to be reached, and appears to recede from the traveller's barque— Ainsi que Dèle sur le mer, gilded like youth, silent as dreams, and like happiness unattainable.' Earlier in the Essay this sensitive writer says:— 'Turner was the first of the Impressionists, and after a lapse of eighty years he remains the greatest, at least in the styles he has treated. That Impressionism came from England is proved by the letters of Delacroix, and demonstrated by M. Paul Signac in his pamphlet on "Neo-Impressionism." ... Turner is the father of the Impressionists. Their discoveries are his. He first saw that Nature is composed in a like degree of colours and of lines, and, in his evolution, the rigid and settled lines of his early method gradually melt away and vanish in the colours. He sought to paint the atmosphere, the envelopment of coloured objects seen at a distance, rather than the things enveloped: and he quickly realised that the atmosphere could not be expressed, except through the infinite parcelling out of things which Claude Lorrain drew in a solid grouping and painting en bloc. He shredded the clouds. He took the massive and admirable masses, the cumuli of Ruysdael, of Hobbema, of Van de Velde, picked the threads out of them, and converted them into a myriad-shaded charpie, which he entrusted to the winds of heaven.'
- 63. Plate XXXV. San Benedetto, Looking Towards Fusina (1843) National Gallery Time has been cruel to both these Venetian pictures, perhaps cruel only to be kind. Even in Ruskin's time much of the transparency had gone; but there they are, dreams of Venice; not the Venice we see, not the Venice that Canaletto saw, but the Venice that floated before the eyes of Turner, that blossomed in the imagination of an old man nearing his seventieth year. I suppose we must call the other pictures of 1843 failures, but only because he tried to express the inexpressible—such themes as 'The Evening of the Deluge' and 'The Morning After,' with Moses writing the book of Genesis, mixed up with Goethe's theory of Light and Colour, and accompanied by an extract from the Fallacies of Hope:— 'The ark stood firm on Ararat: the returning sun Exhaled earth's humid bubbles, and emulous of light, Reflected her lost forms, each in prismatic guise.'
- 64. In this year, too, he exhibited 'The Opening of the Walhalla,' which has been banished to the honourable seclusion of the Dublin National Gallery. This Doric temple, erected on a hill overlooking the Danube, containing two hundred marble busts of eminent Germans, had been opened by King Ludwig of Bavaria in the previous year. The idea inspired Turner; he painted a characteristic picture of the ceremony and sent it to King Ludwig, who returned the gift with the comment that he did not understand it. Poor Turner! Munich would be well content to own the 'Walhalla' now. In 1843 the first volume of Modern Painters was published, which 'originated,' as Ruskin tells us, 'in indignation at the shallow and false criticisms of the periodicals of the day of the works of the great living artist to whom it principally refers.' The second volume was not published until 1846; the third and fourth in 1856, and the fifth and last volume of this 'enormous work of thought, inspiration, sincerity and devotion' in 1860. We have it on the authority of Thornbury, that Turner was vexed at Ruskin's panegyrics, and said, 'The man put things into my head I never thought of.' I doubt if Turner was vexed at the panegyrics, but it is quite certain that Ruskin's imagination saw things in the pictures that Turner never 'thought of.' Turner was a man of deeds, not of thoughts. He worked with his eyes, hand, and spirit: he was Nature's lover. It is certain, too, that after the first irritation felt by his contemporaries at some of the wilder works of Turner's later years had cooled, his fame would have steadily increased, and would have been as high as it is to-day, had Modern Painters never been written.
- 65. Plate XXXVI. The Seelisberg—Moonlight. Water colour (about1843) In the collection of W. G. Rawlinson, Esq. (Size, 11 x 9) Neither that wonderful book, nor any other book, could serve Turner. Only he himself could have produced that fantasy, exquisite and intelligible, called 'The Seelisberg: Moonlight,' or the study, purple, gold and blue, in the Victoria and Albert Museum, of a lake, perhaps Brienz, enclosed by snowy peaks, with the wraith of a castle in the foreground, and the moon in the blue sky. He went his own way, and perhaps on the very day that he should have been reading the glowing periods of Modern Painters, hailing him as a sort of superman, he was the chief actor in that scene on board the old Margate steamer, watching the effect of the sun, and the boiling foam in the wake of the boat, and at luncheon-time eating shrimps out of an immense silk handkerchief laid across his knees. And while he was eating shrimps and watching the movement of the water, those who
- 66. had reached the end of the first volume of Modern Painters were perhaps reading with shining eyes and lifted hearts the concluding passage about 'the great artist whose works have formed the chief subject of this treatise':— 'In all that he says, we believe: in all that he does, we trust.... He stands upon an eminence, from which he looks back over the universe of God, and forward over the generations of men. Let every work of his hand be a history of the one, and a lesson to the other. Let each exertion of his mighty mind be both hymn and prophecy; adoration to the Deity, revelation to mankind.' That is Ruskin at his finest: here is Turner at his—well, as Turner. A Mr. Hammersley, who visited him about this time in Queen Anne Street, described how he heard the shambling, slippered footstep coming down the stairs, the cold, cheerless room, the gallery, even less tidy and more forlorn, all confusion, mouldiness and wretched litter; most of the pictures covered with uncleanly sheets, and the man! 'his loose dress, his ragged hair, his indifferent quiet—all indeed that went to make his physique and some of his mind, but above all I saw, felt (and feel still) his penetrating gray eye.' CHAPTER L 1844: AGED SIXTY-NINE HE EXHIBITS 'RAIN, STEAM AND SPEED,' AND TWICE TRIES TO CROSS THE ALPS ON FOOT The Sketch-Books of 1844 tell the happy story of continental rambles, with flashes of humour, such as this written in pencil against a water- colour of 'Rockets': 'Coming events cast their lights before them.'
- 67. He is at Lucerne, Thun, Interlaken, Lauterbrunnen, Grindelwald, Meiringen, Rheinfelden and Heidelberg and each book has its numerous sketches. To show how unwearyingly this veteran pursued beauty, I quote in full the titles of the drawings in the short 'Lucerne' Sketch-Book, which has not been broken up:— Page 1. Lake and sky. Water-colour. „ 2. do. do. „ 3. do. do. „ 4-9. Blank „ 10. Lake and sky. Water-colour. „ 11. do. Stormy weather. Water-colour. „ 12. The Righi: storm clearing off. Water-colour. „ 13. A Stormy sunset. Water-colour. „ 14. The Rockets. Water-colour. Written in pencil in margin—'Coming events cast their lights before them.' „ 15. The blue Righi. Water-colour. „ 16. The red Righi. „ 17. The rain, with rainbow. Water-colour. „ 18. The rainbow. Water-colour. „ 19. Clearing up a little. Water-colour.
- 68. „ 20. Still raining. Water-colour. „ 21. The rainbow. Water-colour. „ 22. A gleam of sunshine. Water-colour. „ 23. Sunset. Water-colour. „ 24. The Righi. Water-colour. (18 leaves drawn on.) Plate XXXVII. Rain, Steam, and Speed (1844) Tate Gallery The exhibited pictures included that masterpiece in impressionism, 'Rain, Steam, and Speed.' Turner's whole life may be said to have
- 69. been a preparation for this tour de force; all the knowledge that he had acquired, all the facts that he had accumulated, are used in this brilliant synthesis of the effect upon the eye of rushing movement through atmosphere. Has Claude Monet, who acknowledged the impulse he received from studying Turner in 1870, ever visualised movement, light and atmosphere in one impression, as did this wonderful Turner in his seventieth year? But though his power to express a fleeting vision was at its height in this picture, his ability to express his thoughts was as stumbling as ever, shown by the following—printed with other letters by Sir Walter Armstrong in his volume on Turner:— '47 Queen Anne Street, Dec. 28th, 1844. 'Dear Hawkesworth,—First let me say I am very glad to hear Mrs. Fawkes has recovered in health so as to make Torquay air no longer absolute, and that the Isle of Wight will, I do trust, completely establish her health and yours (confound the gout which you work under), tho' thanks to your perseverance in penning what you did, and likewise for the praises of a gossiping letter, thanks to Charlotte Fawkes, who said you thought of Shanklin, but you left me to conjecture solely by the postmark Shanklin—Ryde— so now I scribble this to the first place in the hope of thanking your kindness in the remembrance of me by the Yorkshire Pie equal good to the olden time of Hannah's culinary exploits. 'Now for myself, the rigours of winter begin to tell upon me, rough and cold, and more acted upon by changes of weather than when we used to trot about at Farnley, but it must be borne with all the thanks due for such a lengthened period. 'I went, however, to Lucerne and Switzerland, little thinking of supposing such a cauldron of squabbling, political or religious, I was walking over. The rains came on early so I could not cross the Alps, twice I tried, was sent back with a wet jacket and worn- out boots, and after getting them heel-tapped, I marched up
- 70. some of the small valleys of the Rhine and found them more interesting than I expected. 'Now do you keep your promise and so recollect that London is not so much out of nearest route to Farnley now ... Shanklin, and (I) do feel confoundedly mortified in not knowing your location when I was once so near you, for I saw Louis Philippe land at Portsmouth.—Believe me, dear Hawkesworth, Yours most sincerely, 'J. M. W. Turner.' Another blow fell upon Turner this year. The Mr. Hammersley aforementioned visited him again in Queen Anne Street, and gives the following account:— 'Our proceedings resembled our proceedings on the former visit, distinguished from it, however, by the exceeding taciturnity, yet restlessness of my great companion, who walked about and occasionally clutched a letter which he held in his hand. I feared to break the dead silence, varied only by the slippered scrape of Turner's feet, as he paced from end to end of the dim and dusty apartment. At last he stood abruptly, and turning to me, said, "Mr. Hammersley, you must excuse me, I cannot stay another moment; the letter I hold in my hand has just been given to me, and it announces the death of my friend Callcott." He said no more; I saw his fine gray eyes fill as he vanished, and I left at once.' The loss of friends set his mind dwelling upon the past, and it was no doubt in gratitude to all he owed to Ruysdael that he painted and exhibited this year the vivacious sea-piece now in the National Gallery, which he called 'Fishing-Boats Bringing a Disabled Ship into Port Ruysdael.' Needless to say, there is no such port anywhere. He also exhibited the beautiful Approach to Venice' in the possession of Sir Charles Tennant; and—the old man twice tried to cross the Alps on foot, referred to in the above letter, which is almost as wonderful as painting a picture. It would seem that he really succeeded in the
- 71. enterprise if 'passed' means 'crossed,' as in the 'Grindelwald' Sketch- Book, against a drawing of mountains, is the following scrawl:— 'No matter what bef [? befell] Hannibel—W.B. and J.M.W.T. passed the Alps from [? near] Fombey [?] Sep. 3, 1844.' CHAPTER LI 1845: AGED SEVENTY PICTURES OF WHALERS, AND AN ENTRY ON THE LAST PAGE OF HIS LAST SKETCH-BOOK Now, when he is nearing his decline, Turner is described as stooping very much, and looking down. Thinking of Turner 'looking down,' I recall the story that came to Sir Walter Armstrong from Mr. Stopford Brooke: how some one who knew Turner, at least by sight, was one day passing along the wharves beyond the Palace of Westminster, when he noticed the figure of a sturdy man in black squatting on his heels at the river's edge, and looking down intently into the water. Passing on, he thought for the moment no more about it. But on his return, half an hour later, the figure was still there, and still intent in the same way. That watcher was Turner, and the object of his interest was the pattern made by the ripples at the edge of the tide. Ruskin says that this year his health, and with it in great degree, his mind, failed suddenly. And to Ruskin we owe this pathetic passage:— 'The last drawing in which there remained a reflection of his expiring power, he made in striving to realise, for me, one of these faint and fair visions of the morning mist fading from the Lake of Lucerne. '"There ariseth a little cloud out of the sea like a man's hand ... For what is your life?"'
- 72. Plate XXXVIII. Sunrise With a Sea Monster (about 1845) Tate Gallery And Turner was going his own way, making his little jokes. On June 31st, 1845, he wrote to Mr. E. Bicknell of Heme Hill:— 'My Dear, SIR,—I will thank you to call in Queen Anne Street at your earliest convenience, for I have a whale or two on the canvas.' This letter, of course, referred to the 'Whalers' pictures, exhibited in 1845 and 1846. The 'Whalers' Sketch-Book contains drawings of 'Steamer Leaving Harbour,' 'Burning Blubber,' 'Whalers at Sea,' 'Study of Fish,' etc. Perhaps he made a voyage; perhaps he talked with sailors in one of his haunts at Wapping, and learnt from them of the wonders of the
- 73. deep waters related by Arctic voyagers. However the idea or the vision came he now makes sketches of whaling subjects and paints pictures of 'Whalers,' one of which is in the Turner Gallery, four boats' crews attacking their prey with harpoons, and in the background are the white sails of their vessels, dimly seen through mists and snow wreaths. The imaginative 'Sunrise with a Sea Monster' probably belongs to the 'Whalers' period. On the misty waters of the ocean, reflecting a yellow sunrise, a sea monster, with a head like a magnified red gurnet, advances, the huge head towering out of the water. In the distance are forms suggesting icebergs. Punch had a genial sneer at a 'Whalers' picture:— 'It embodies one of those singular effects which are only met with in lobster salads and in this artist's pictures. Whether he calls his picture "Whalers" or "Venice," or "Morning," or "Noon," or "Night," it is all the same; for it is quite as easy to fancy it one thing as another.' Thornbury is responsible for the following:— 'I am afraid the tradition is too true, that that great and bitter satirist of poor humanity's weaknesses, Mr. Thackeray, had more than a finger in thus lashing the dotage of a great man's genius. Long after, I have heard that Mr. Thackeray was shown some of Turner's finest water-colour drawings, upon which he exclaimed: "I will never run down Turner again." But the blows had already gone to the old man's heart, and it did no good to lament them then.' In the Sketch-Books of 1845 and 1846, we find him at 'Folkestone,' 'Hythe and Walmer,' 'Ambleteuse and Wimereux,' 'Boulogne,' 'Eu and Treport,' 'Dieppe,' and back again at 'Folkestone.' In the last of all the Sketch-Books, 'Kent,' 1845-46, when Turner was over seventy, is this against a drawing of 'Houses and Church': 'May 30. Margate, a small opening along the horizon marked the approach of the sun by its getting yellow,' etc.
- 74. Plate XXXIX. Tell's Chapel, Fluelen. Water colour (1845) In the collection of W. G. Rawlinson, Esq. (Size, 11 3/8 x 9) A little later in this valedictory Sketch-Book is the following in his own handwriting:— 'May. Blossoms. Apple, Cherry, Lilac, Small white flowers in the Hedges, in Clusters, D. Blue Bells, Buttercups and daisies in the fields, Oak, Warm, Elm G., Ash, yellow,' etc. With that utterance of joy in nature we may take our leave of the Sketch-Books, and of the close of the great period of Turner, thinking of small white flowers in the hedges, buttercups and daisies in the
- 75. fields, seen by his old eyes, and recorded tremblingly in his last Sketch-Book. There is no sign of trembling in the exquisite vision of 'Tell's Chapel—Fluelen,' his adieu to Switzerland, perhaps the last water-colour from his hand. PART EIGHT 1846-1851 THE YEARS OF DECLINE—AND THE END CHAPTER LII 1846: AGED SEVENTY-ONE THE BEGINNING OF TURNER'S DECLINE, AND A 'GREY, DIM DRAWING' The story of Turner's art life really ended in the last chapter: there is little more to tell, yet 'Queen Mab's Grotto,' which he exhibited at the British Institution in 1846, flickers with the old splendour. The sultry arch of trees in the foreground, the golden castle rising to the sky, have something of the old witchery, and the mundane fairies are more attractive than many of his clothed foreground fishermen. In this picture he rivalled nobody but himself, but the suggestion clearly came from Shakespeare, and it was the old man's pleasure to couple the names of Shakespeare and Turner in the catalogue, with this from A Midsummer Night's Dream:— 'Frisk it, frisk it by the moonlight beam.' And this from the Fallacies of Hope:—
- 76. 'Thy orgies, Mab, are manifold.' The other pictures of this year have the old extravagance of title, little more. They were:— 'Hurrah for the Whaler Erebus! Another Fish!' 'Undine giving the Ring to Massaniello, Fisherman of Naples.' 'The Angel Standing in the Sun,' with quotations from Revelation and the poet Rogers. 'Whalers (boiling blubber) entangled in flaw ice, endeavouring to extricate themselves.' 'Returning from the Ball (St. Martha).' 'Going to the Ball (San Martino).' His ambition was as buoyant as ever, and the look of his eyes as keen; but his hand was beginning to lose its power. Ruskin has this curt comment:— 'I shall take no notice of the three pictures painted in the period of his decline ("Undine," "The Angel Standing in the Sun," and "The Hero of a Hundred Fights"). It was ill-judged to exhibit them; they occupy to Turner's other works precisely the relation which Count Robert of Paris and Castle Dangerous hold to Scott's early novels.' One could continue indefinitely quoting Ruskin on Turner, ranging, as he does, through the whole gamut from eulogy to chastisement, from adoration to grief. Here is a passage that arrests me as I turn his pages: pathetic, but a wilful misunderstanding of Turner's temperament:— 'There is something very strange and sorrowful in the way Turner used to hint only at these under-meanings of his; leaving us to find them out, helplessly; and if we did not find them out, no word more ever came from him. Down to the grave he went,
- 77. silent. "You cannot read me; you do not care for me; let it all pass; go your ways."' Plate XL. Queen Mab's Grotto (1846) National Gallery And here is a wail that is probably quite within the sad truth. In a note to the first volume of Modern Painters, after remarking sadly that 'Turner is exceedingly unequal,' that he has failed most frequently 'in elaborate compositions,' and that 'finding fault with Turner is not either decorous in myself or likely to be beneficial to the reader,' Ruskin continues:— 'The reader will have observed that I strictly limited the perfection of Turner's works to the time of their first appearing on the walls of the Royal Academy. It bitterly grieves me to have to do this, but the fact is indeed so. No picture of Turner's is seen in
- 78. perfection a month after it is painted. The 'Walhalla' cracked before it had been eight days in the Academy rooms; the vermilions frequently lose lustre long before the Exhibition is over; and when all the colours begin to get hard a year or two after the picture is painted, a painful deadness and opacity come over them, the whites especially becoming lifeless, and many of the warmer passages settling into a hard valueless brown, even if the paint remains perfectly firm, which is far from being always the case. I believe that in some measure these results are unavoidable, the colours being so peculiarly blended and mingled in Turner's present manner, as almost to necessitate their irregular drying; but that they are not necessary to the extent in which they sometimes take place, is proved by the comparative safety of some even of the more brilliant works. Thus the "Old Téméraire" is nearly safe in colour, and quite firm; while the "Juliet and Her Nurse" is now the ghost of what it was; the "Slaver" shows no cracks, though it is chilled in some of the darker passages, while the "Walhalla" and several of the recent Venices cracked in the Royal Academy.' How the attacks and parodies of Turner must have pained Ruskin! This, for example, from Punch on 'Venice, Morning, Returning from the Ball':— 'We had almost forgotten Mr. J. M. W. Turner, R.A., and his celebrated MS. poem, the Fallacies of Hope, to which he constantly refers us, as "in former years"; but on this occasion, he has obliged us by simply mentioning the title of the poem, without troubling us with an extract. We will, however, supply a motto to his "Morning—Returning from the Ball," which really seems to need a little explanation; and as he is too modest to quote the Fallacies of Hope, we will quote for him:— 'Oh, what a scene! Can this be Venice? No. And yet methinks it is—because I see Amid the lumps of yellow, red and blue, Something which looks like a Venetian spire.
- 79. That dash of orange in the background there Bespeaks 'tis morning. And that little boat (Almost the colour of Tomato sauce) Proclaims them now returning from the ball: This is my picture I would fain convey, I hope I do. Alas! what Fallacy!' Plate XLI. Lake With Distant Headland and Palaces. Water colour (1840 or after) Tate Gallery The following pen-picture is no parody. Wilkie Collins told Thornbury that when a boy— 'He used to attend his father on varnishing days, and remembers seeing Turner (not the more perfect in his balance for the brown sherry at the Academy lunch), seated on the top of a flight of steps, astride a box. There he sat, a shabby Bacchus, nodding like a Mandarin at his picture, which he, with a pendulum motion,
- 80. now touched with his brush, and now receded from. Yet, in spite of sherry, precarious seat and old age, he went on shaping in some wonderful dream of colour; every touch meaning something, every pin's head of colour being a note in the chromatic scale.' There is nothing sad in that; but who can look at or recall that 'grey, dim drawing, with one or two specks of light from craft on the river,' called 'Twilight in the Lorreli,' without emotion? This was one of the fifty-three drawings that Turner had brought years before straight to Farnley on his return from the Rhine. Long afterwards, possibly in this year, Hawkesworth Fawkes conveyed the set to the dreary house in Queen Anne Street to show to their creator. The old man turned over the drawings until he came to 'Twilight in the Lorreli.' His eyes filled with tears, and he muttered, 'But, Hawkey! but, Hawkey!' CHAPTER LIII 1847, 1848 AND 1849: AGED SEVENTY-TWO TO SEVENTY-FOUR HE DISAPPEARS FROM HIS OLD HAUNTS, AND IS INTERESTED IN OPTICS AND PHOTOGRAPHY Turner's art life almost ceased during the years 1847, 1848 and 1849. Three old-new pictures only were exhibited: 'The Hero of a Hundred Fights,' probably an early picture re-touched, and two works of former years: 'The Wreck Buoy,' which he repainted, spending 'six laborious days' upon it, and 'Venus and Adonis,' dating from nearly fifty years before, after his visit to the Louvre in 1802. The interest of these years, if it be an interest, is centred in his cunning and successful efforts to escape from the notice of friends and companions, and to withdraw his private life from any kind of intrusion. The doors of Queen Anne Street were locked and barred,
- 81. and when he was absent from home, which was often, his old housekeeper, Hannah Danby, had no knowledge of his hiding-place. Sometimes he was seen at a council meeting of the Royal Academy or on Varnishing Day, but his friends were rarely able to obtain speech with him. Hawkesworth Fawkes tried to keep up acquaintance with his father's old Mend, and every Christmas a hamper arrived in Queen Anne Street from Farnley. There is a letter to Hawkesworth dated December 27th, 1847, beginning:—' Many thanks for the P.P.P., viz., Pie, Phea, and Pud—the Xmas cheer in Queen Anne Street.' One day, so the story runs, an artist took shelter in a public-house, where he found Turner sitting in the furthest corner with his glass of grog before him. Said the unnamed artist: 'I didn't know you used this house. I shall often drop in now I know where you quarter.' Turner emptied his glass, and as he went out said, 'Will you? I don't think you will.' The secret of his hiding-place was not discovered until a day or two before his death. As everybody now knows, he lived mainly, during those last years, in the little house with the roof balcony facing the Thames at Cremorne, in what is called to-day Cheyne Walk. The story current for years was that he passed the house in one of his rambles, saw that rooms were vacant, liked the place, and after some bargaining with the landlady, agreed co become the tenant. He asked her name, and upon receiving the answer, 'Mrs. Booth,' chuckled, 'Then I'll be Mr. Booth.' This story is incorrect, as he had made the acquaintance of Sophia Caroline Booth years before, when she let lodgings at Margate. As it is believed that Turner paid his last visit to Margate in 1845, it is probable that he transferred Mrs. Booth to the little house at Chelsea in that year. Her name appears in a codicil to his will, dated February 1st, 1849, giving her the same provision as Hannah Danby, his housekeeper in Queen Anne Street, who had entered his service, a girl of sixteen, in the year 1801. Hannah Danby and Mrs. Booth both survived him.
- 82. Turner's curiosity, his eagerness for wider knowledge about his art and all that pertained to it, never relaxed, even in this period of his failing powers. One of the most interesting chapters in Thornbury's Life is the account given by Mayall, the photographer of Regent Street, of Turner's interest in optics and photography. I append portions of the information furnished by Mayall, whom Thornbury describes as 'that eminent professor in the progressing and wonderful art':— 'Turner's visits to my atelier were in 1847, 1848 and 1849. I took several admirable daguerreotype portraits of him, one of which was reading, a position rather favourable for him on account of his weak eyes and their being rather bloodshot.... My first interviews with him were rather mysterious; he either did state, or at least led me to believe, that he was a Master in Chancery, and his subsequent visits and conversation rather confirmed this idea. At first he was very desirous of trying curious effects of light let in on the figure from a high position, and he himself sat for the studies.... He stayed with me some three hours, talking about light and its curious effects on films of prepared silver. He expressed a wish to see the spectral image copied, and asked me if I had ever repeated Mrs. Somerville's experiment of magnetising a needle in the rays of the spectrum. I told him I had. 'I was not then aware that the inquisitive old man was Turner, the painter. At the same time, I was much impressed with his inquisitive disposition, and I carefully explained to him all I then knew of the operation of light on iodized silver plates. He came again and again, always with some new notion about light....' Mayall tells us that Turner when he visited him bore the marks of age; but in the profile drawing of this period, ascribed to Linnell, with the straggling hair, the powerful nose, and the enormous stock about his neck, the face is keen, and the artist has quite caught the gleam of the grey eye. This drawing is not by Linnell, as has been hitherto supposed, but by Landseer and Count d'Orsay in conjunction. Mr. A.
- 83. S. Bicknell, who was present when the sketch was made, contributed to the Athenæum of January 9th, 1909, the following letter on this subject, as interesting as it is authoritative:—' 'A few days ago I first saw a handsome quarto "Turner, by Sir Walter Armstrong, 1902," in which, as a second frontispiece, I found a head and shoulders portrait of that great artist, described on the opposite leaf as "from the sketch in water-colours by J. Linnell, in the collection of James Orrock, Esq." 'During the last fifty years I have occasionally come across a reference to this likeness, declaring that it was probably the work of some contemporary painter, sketched at a meeting or private entertainment; but as these surmises have at length crystallised into a positive assertion concerning Linnell, I think it may be well to place the truth on record. 'My father, Elhanan Bicknell, of Herne Hill, frequently entertained at dinner a large company of the most distinguished artists and patrons of art, amongst whom Turner, but never Linnell, was often one. It being the case that Turner objected to having his portrait taken, on an occasion of that kind two conspirators, Count D'Orsay and Sir Edwin Landseer, devised a little plot to defeat the result of this antipathy. Whilst Turner unsuspiciously chatted with a guest over a cup of tea in the drawing-room, D'Orsay placed himself as screen beside him to hide, when necessary, Landseer sketching him at full length in pencil on the back of a letter. Landseer gave what he had done to D'Orsay, who, after re-drawing it at home, and enlarging the figure to eight inches in height, sold it to J. Hogarth, printseller in the Haymarket, for twenty guineas; and it was then lithographed and published by the latter, January 1st, 1851, with the title of Turner's mysterious poem, The Fallacies of Hope, at the bottom. Sixteen copies were included in the Bicknell sale at Christie's in 1863. The Louis XIV. panelling of the room, as well as a piano, inlaid with Sèvres plaques, are indicated in the background; and I may also mention that I was present at that party, which took
- 84. place, to the best of my belief, about Christmas, 1847, or early in 1849. 'Ruskin, who seldom admitted any blemish, even in the person of his hero, called this portrait a caricature, but it was nothing of the kind; I knew Turner extremely well, and I have always considered it to be a most admirable, truthful likeness; indeed, the only one exactly portraying his general appearance and expression in his latter years.' So here we have a likeness of Turner in the period of his decline and disappearance from his old haunts, an authentic likeness at one of his re-appearances—chatting with a guest over a cup of tea in a Herne Hill drawing-room. CHAPTER LIV 1850: AGED SEVENTY-FIVE HIS LAST FOUR PICTURES PAINTED IN HIDING AT CHELSEA In 1850, the year before his death, Turner sent four pictures to the Royal Academy, an heroic attempt on the veteran's part to assure the world that his power had not deserted him; but these canvases are but the tottering ruins of his genius, and they have been hung among other 'splendid failures' in the large, lower room of the Turner Gallery. But, as I have said before, Turner's 'splendid failures' are merely less great than his triumphs. His 'failures' in the large, lower room of the Turner Gallery, would easily make a lesser man's reputation. These four valedictory works entitled 'Æneas relating his Story to Dido,' 'Mercury sent to admonish Æneas,' 'The Departure of the Trojan Fleet,' and 'The Visit to the Tomb,' were painted between January and April, 1850, in a small room, with a small window, in the little house at Cremorne. We are told that at this window, and on the roof
- 85. balcony, he would spend a long time each day studying the ways of the sun, the effect of light on the river and on the open places of rural Chelsea; and that he would often rise early, paint for a little, and then return to bed. Mrs. Booth declared that some of his last work was inspired by his dreams; that one night she heard him calling out excitedly; that she gave him the drawing materials he asked for, and that he made some notes, which he afterwards used for a picture. Mrs. Booth also confessed that she could not resist whispering in the neighbourhood that 'Booth' was a great man in disguise, and that when he died he would surely be buried in St. Paul's. This local gossip was collected later by John Pye the engraver. Here I may print, for what it is worth, a letter, that was sent to me by an unknown correspondent in reference to a small book on Turner I wrote three years ago:— 'Clapham, March 1907. 'Re Turner. 'Dear Sir,—In the eighties (I think) there resided at Haddenham Hall, Haddenham, Bucks, a Mrs. Booth, whom it was understood was Turner's widow. I expressed a wish to look over the Hall, and was received by the old lady herself (she was a very homely body, and always wore a big cotton apron). In one of the rooms I recognised a miniature portrait of the late Dr. Price of Margate. Mrs. Booth said, "Yes! it was painted by my husband, Mr. Turner the artist; he and the Doctor were great friends." I also understood that Turner lodged with her when painting his pictures of Margate. 'When Mrs. Booth died she was taken to Margate to be buried. As I have never read of Turner's marriage, this may prove interesting. 'P.S.—The son of the late Dr. Price still resides at Margate.' In a letter to Hawkesworth Fawkes, dated December 27th, 1850, Turner wrote: 'Old Time has made sad work with me since I saw you
- 86. in town.' But a certain dinner at David Roberts's house shows that old Time did not prevent him from being merry and sociable after his manner. The account of this dinner in 1850 is printed in a note to Ballantyne's Life of David Roberts. Turner's manner at the feast is described as— 'Very agreeable, his quick bright eye sparkled, and his whole countenance showed a desire to please. He was constantly making, or trying to make jokes; his dress, though rather old- fashioned, was far from being shabby. Turner's health was proposed by an Irish gentleman who had attended his lectures on perspective, on which he complimented the artist. Turner made a short reply in a jocular way, and concluded by saying, rather sarcastically, that he was glad this honourable gentleman had profited so much by his lectures as thoroughly to understand perspective, for it was more than he did. Turner afterwards, in Roberts's absence, took the chair, and, at Stanfield's request, proposed Roberts's health, which he did, speaking hurriedly, but soon ran short of words and breath, and dropped down on his chair with a hearty laugh, starting up again and finishing with a "hip, hip, hurrah!"... Turner was the last who left, and Roberts accompanied him along the street to hail a cab.... When the cab drove up, he assisted Turner to his seat, shut the door, and asked where he should tell cabby to take him; but Turner was not to be caught, and, with a knowing wink, replied, "Tell him to drive to Oxford Street, and then I'll direct him where to go."' Sir Martin Shee died this year, and it is said that Turner was aggrieved that he was not offered the Presidentship of the Royal Academy. It is difficult to realise Turner in that office this year or in any year of his life. He was not made for official duties, but to make beautiful and wonderful things. CHAPTER LV
- 87. 1851: AGED SEVENTY-SIX THE MYSTERY OF THE LAST YEARS OF HIS LIFE REVEALED TO HIS FRIENDS: AND HIS DEATH I leaned against the parapet of the Embankment in Cheyne Walk, Chelsea, and gazed at the row of cosy little houses on the other side of the road that face the Thames. The house where Turner died, I had been told, is now 119 Cheyne Walk. My eyes sought 119, but found it not. The numbers passed from 118 to 120. Then I crossed the road to discover that Nos. 118 and 119 have been converted into one house. Peering, I discerned, almost hidden by Virginia creeper, a tablet saying that here Turner died. So this was the house. Somewhere near here 'Puggy Booth,' as he was known to the street boys, 'Admiral Booth' to the tradesmen, moored his boat. The story was current in Chelsea that he was an Admiral in reduced circumstances, and Turner was not the man to illumine a mystery, or end a joke. We learn from Thornbury that up to the period of his final illness, he would often rise at daybreak, leave his bed with some blanket or dressing-gown carelessly thrown over him, and ascend to the railed-in roof to watch the sunrise, and see the colour flush the morning sky.
- 88. Plate XLII. Lake of Brienz. Water colour (about 1843) Victoria and Albert Museum There was the railed-in roof, crowning the 'Cremorne Cottage,' that in Turner's time had green sward to the edge of the river: the house with three windows only, one in the basement, and one each on the first and second floors. In the room on the second floor, where he painted his last four pictures, he died. I remembered what I had read of the talk of the undertaker's men about the shabbiness of the place, and the narrowness of the staircase, so circumscribed, that to carry the coffin up was impossible: they were obliged to convey the body down to the coffin. Then my thoughts turned to Turner the artist, the poet in paint, and I recalled what his great contemporary, Constable, had said of him: that one of Turner's early pictures, 'a canal with numerous boats making thousands of beautiful shapes,' was 'the most complete work of genius' he had ever seen; that 'Turner's light, whether it emanates
- 89. from sun or moon, is exquisite'; that 'he seems to paint with tinted steam, so evanescent and so airy'; and then I repeated the passage about the golden visions glorious and beautiful, only visions, but pictures to live and die with. So I mused, turning from that sad little house, now so cheerful, to gaze upon the Thames beloved by Turner. He was born near the river; he chose his rural retreats at Hammersmith and Twickenham because they were by the banks; and Wapping was the scene of his later jaunts. Almost his first oil picture, 'Moonlight at Millbank,' was painted by the riverside; one of his earliest drawings was 'The Archbishop's Palace at Lambeth.' I rarely pass the wharves south of the Houses of Parliament without seeing him, as in a vision, squatting on his heels, and gazing for half an hour at a time at the ripples. The magnificent new home of his pictures is by the Thames at Millbank, and his last journey but one was from the Thames: his last journey was to the crypt of St. Paul's on the hill above the river: there he was rendered to the mould:— 'Under the cross of gold That shines over city and river, There he shall rest for ever Among the wise and the bold.' There, in the crypt, he was buried as he desired, by the side of Sir Joshua Reynolds, and his funeral, as he desired and stipulated in his will, cost one thousand pounds. When I returned home from musing before the Turner Cottage, I re- read the story of the last years of his life, how his hiding-place was discovered, and so on to the end, and after. The true facts were revealed through the pertinacity of John Pye the engraver, who 'left certain memoranda of events connected with "Admiral Booth's" tenancy of the Cremorne Cottage, and death under its roof, which are of extraordinary interest.' Pye's memoranda were summarised by Sir Walter Armstrong in his volume on Turner, partly from a copy made by the late Sir Frederic Burton, and partly from information supplied to Sir Walter by Mr. J. L. Roget, through whose hands the whole of Pye's manuscripts passed.
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