![Introduction
To a large extent this book is an updated version of Lectures on Topological
Dynamics by Robert Ellis [Ellis, R., (1969)]. That book gave an exposition
of what might be called an algebraic theory of minimal sets. Our goal here
is to give a clear, self contained exposition of a new approach to the theory
which allows for more straightforward proofs and develops a clearer language
for expressing many of the fundamental ideas. We have included a treatment
of many of the results in the aforementioned exposition, in addition to more
recent developments in the theory; we have not attempted, however, to give
a complete or exhaustive treatment of all the known results in the algebraic
theory of minimal sets. Our hope is that the reader will be motivated to use the
language and techniques to study related topics not touched on here. Some of
these are mentioned either in the exercises or notes given at the end of various
sections. This book should be suitable for a graduate course in topological
dynamics whose prerequisites need only include some background in topology.
We assume the reader is familiar with compact Hausdorff spaces, convergence
of nets, etc., and perhaps has had some exposure to uniform structures and
pseudo metrics which play a limited role in our exposition.
A flow is a triple (X, T, π) where X is a compact Hausdorff space, T is a
topological group, and π : X × T → X is a continuous action of T on X,
so that xe = x and (xt)s = x(ts) for all x ∈ X, s, t ∈ T . Here we write
xt = π(x, t) for all x ∈ X and t ∈ T , and e is the identity of the group
T . Usually the symbol π will be omitted and the flow (X, T, π) denoted by
(X, T ) or simply by X. In the situations considered here there is no loss of
generality if T is given the discrete topology. The assumptions made thus far
do not suffice to produce an interesting theory. The group T may be too “small”
in its action on X. Thus for example, the trivial case where xt = x0, a fixed
element of X, for all x ∈ X and t ∈ T , is not ruled out. To eliminate such
degenerate behavior it is convenient to assume that the flow (X, T ) is point
ix](https://image.slidesharecdn.com/2352940-250527142201-702e6437/85/Automorphisms-And-Equivalence-Relations-In-Topological-Dynamics-David-B-Ellis-15-320.jpg)
![2
Flows and their enveloping semigroups
For us a flow will be a compact Hausdorff space X provided with a continuous
(right) action of a group T on X. In Topological Dynamics we are concerned
with the so-called asymptotic behavior of this action. This motivates the con-
sideration of not just the collection T thought of as a subset of the set XX
of self-mappings of X, but all of the limit points of T in XX. This gives rise
to the notion of the enveloping semigroup E(X, T ) of the flow. The compo-
sition of functions gives a natural semigroup structure on E(X, T ) which as
we will see in subsequent sections, can be exploited to study the asymptotics
of the original flow (X, T ). The semigroup structure on βT discussed in the
previous section and its appendix serves as a prototype example. Indeed since
T ⊂ βT , this semigroup structure makes (βT, T ) itself a flow, and we will see
in proposition 2.9 that E(βT, T ) ∼
= βT in a natural way.
In this section we introduce the appropriate notation, the details of the con-
struction, and give an exposition of some of the elementary properties of
E(X, T ). Many of these properties will be used directly, and serve as moti-
vation in what follows. Several accounts of this material appear in the text-
book-literature (see for example [Auslander, J., (1988)] and [Ellis, R., (1969)].
However, the point of view we have adopted here is slightly different. In order
to emphasize the connection between the two, we have involved βT in the
definition of E(X, T ) (see 2.8). In particular the fact that E(X, T ) is a homo-
morphic image of βT , both as a flow and as a semigroup has important conse-
quences. We begin with some basic notation and definitions.
Definition 2.1 Let X be a set and S a semigroup. Then an action of S on X is
a function
π : X × S → X
(x, s) → xs such that x(st) = (xs)t
19](https://image.slidesharecdn.com/2352940-250527142201-702e6437/85/Automorphisms-And-Equivalence-Relations-In-Topological-Dynamics-David-B-Ellis-39-320.jpg)
![26 Flows and their enveloping semigroups
Exercise 2.2 (see 2.12) Let:
(i) (X, T ) be a flow,
(ii) x0 ∈ X with x0T = X (so that (X, T ) is point transitive), and
(iii) f : E(X) → X be defined by f (p) = x0p for all p ∈ E(X).
Then f induces an isomorphism, θ : E(E(X)) ∼
= E(X).
Exercise 2.3 Let:
(i) E be a compact Hausdorff space provided with a semigroup structure,
(ii) Lp : E → E
m → pm
for all p ∈ E,
(iii) Rp : E → E
m → mp
for all p ∈ E, and
(iv) ϕ : E → EE be defined by ϕ(p) = Rp for all p ∈ E(X).
Then
(a) ϕ is a semigroup homomorphism, and
(b) ϕ is continuous if and only if Lp is continuous for all p ∈ E. In this
case ϕ identifies E with a subsemigroup of EE, and E is referred to as an
E-semigroup (see also [Akin, E. (1997)]). It is immediate from 2.7, that for
any flow (X, T ), its enveloping semigroup E(X, T ) is an E-semigroup.](https://image.slidesharecdn.com/2352940-250527142201-702e6437/85/Automorphisms-And-Equivalence-Relations-In-Topological-Dynamics-David-B-Ellis-46-320.jpg)
![Minimal sets and minimal right ideals 33
6. I =
{Iv | v ∈ J}. (by 1, 5)
7. Finally let u, v ∈ J and p ∈ Iv ∩ Iu.
8. p = pu = pv and there exists q ∈ Iv with qp = v. (by 8, (c))
9. u =
(b)
vu = (qp)u = q(pu) =
8
qp =
8
v.
(e) This is just a restatement of (d).
Note that each of the groups Iv in proposition 3.12 is isomorphic to the group
G = Iu; indeed the map pu → pv is an isomorphism since (pv)(qv) =
p(vq)v = (pq)v. It is interesting to note that given an abstract group G and
an index set J, one can define a semigroup structure on the disjoint union
I =
v∈J Gv of copies of G, in which J can be identified with the set of
idempotents. We simply define pvqw = (pq)w ∈ Gw for p, q ∈ G and
v, w ∈ J. Then {ev | v ∈ J} is the set of idempotents in I. Proposition 3.12
shows that every minimal ideal in E(X, T ) has this structure.
For any semigroup E one can define an equivalence relation on the set J of
idempotents in E as follows:
u ∼ v ⇐⇒ uv = u and vu = v.
This relation is clearly reflexive and symmetric; to check transitivity we observe
that:
u ∼ v ∼ w ⇒ uw = (uv)w = u(vw) = uv = u
and wu = (wv)u = w(vu) = wv = w.
This motivates the following definition.
Definition 3.13 Let (X, T ) be a flow. An idempotent u2 = u ∈ E(X, T )
is said to be a minimal idempotent if u is contained in some minimal ideal
I ⊂ E(X, T ). If u, v ∈ E(X, T ) are idempotents with uv = u and vu = v,
we write u ∼ v and refer to u and v as equivalent idempotents.
When u and v are in the same minimal ideal I, then u ∼ v ⇒ u = v
since by 3.12 both u and v act as left-identities on I. Thus for any minimal
idempotent, the equivalence class [u] intersects each minimal ideal at most
once; we leave it as an exercise for the reader to check that [u] contains only
minimal idempotents. The following proposition says that [u] intersects every
minimal ideal exactly once, so that [u] consists of one idempotent from every
minimal ideal in E(X, T ).
Proposition 3.14 Let:
(i) (X, T ) be a flow,
(ii) E = E(X, T ),](https://image.slidesharecdn.com/2352940-250527142201-702e6437/85/Automorphisms-And-Equivalence-Relations-In-Topological-Dynamics-David-B-Ellis-53-320.jpg)
Automorphisms And Equivalence Relations In Topological Dynamics David B Ellis
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- 7. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 287 Topics on Riemann surfaces and Fuchsian groups, E. BUJALANCE, A.F. COSTA & E. MARTÍNEZ (eds) 288 Surveys in combinatorics, 2001, J.W.P. HIRSCHFELD (ed) 289 Aspects of Sobolev-type inequalities, L. SALOFF-COSTE 290 Quantum groups and Lie theory, A. PRESSLEY (ed) 291 Tits buildings and the model theory of groups, K. TENT (ed) 292 A quantum groups primer, S. MAJID 293 Second order partial differential equations in Hilbert spaces, G. DA PRATO & J. ZABCZYK 294 Introduction to operator space theory, G. PISIER 295 Geometry and integrability, L. MASON & Y. NUTKU (eds) 296 Lectures on invariant theory, I. DOLGACHEV 297 The homotopy category of simply connected 4-manifolds, H.-J. BAUES 298 Higher operads, higher categories, T. LEINSTER (ed) 299 Kleinian groups and hyperbolic 3-manifolds, Y. KOMORI, V. MARKOVIC & C. SERIES (eds) 300 Introduction to Möbius differential geometry, U. HERTRICH-JEROMIN 301 Stable modules and the D(2)-problem, F.E.A. JOHNSON 302 Discrete and continuous nonlinear Schrödinger systems, M.J. ABLOWITZ, B. PRINARI & A.D. TRUBATCH 303 Number theory and algebraic geometry, M. REID & A. SKOROBOGATOV (eds) 304 Groups St Andrews 2001 in Oxford I, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 305 Groups St Andrews 2001 in Oxford II, C.M. CAMPBELL, E.F. ROBERTSON & G.C. SMITH (eds) 306 Geometric mechanics and symmetry, J. MONTALDI & T. RATIU (eds) 307 Surveys in combinatorics 2003, C.D. WENSLEY (ed.) 308 Topology, geometry and quantum field theory, U.L. TILLMANN (ed) 309 Corings and comodules, T. BRZEZINSKI & R. WISBAUER 310 Topics in dynamics and ergodic theory, S. BEZUGLYI & S. KOLYADA (eds) 311 Groups: topological, combinatorial and arithmetic aspects, T.W. MÜLLER (ed) 312 Foundations of computational mathematics, Minneapolis 2002, F. CUCKER et al. (eds) 313 Transcendental aspects of algebraic cycles, S. MÜLLER-STACH & C. PETERS (eds) 314 Spectral generalizations of line graphs, D. CVETKOVIC, P. ROWLINSON & S. SIMIC 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁR (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds)
- 8. 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. 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- 9. London Mathematical Society Lecture Note Series: 412 Automorphisms and Equivalence Relations in Topological Dynamics DAVID B. ELLIS Beloit College, Wisconsin ROBERT ELLIS University of Minnesota
- 10. University Printing House, Cambridge CB2 8BS, United Kingdom Published in the United States of America by Cambridge University Press, New York Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107633223 © D. B. Ellis and R. Ellis 2014 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2014 Printed and bound in the United Kingdom by CPI Group Ltd, Croydon CR0 4YY A catalogue record for this publication is available from the British Library Library of Congress Cataloging-in-Publication data Ellis, D. (David), 1958– Automorphisms and equivalence relations in topological dynamics / David B. Ellis, Beloit College, Wisconsin, Robert Ellis, University of Minnesota. pages cm. – (London Mathematical Society lecture note series ; 412) ISBN 978-1-107-63322-3 (pbk.) 1. Topological dynamics. 2. Algebraic topology. 3. Automorphisms. I. Ellis, Robert, 1926–2013 II. Title. QA611.5.E394 2014 512’.55–dc23 2013043992 ISBN 978-1-107-63322-3 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.
- 11. To Valerie, Carrie, and Kathleen; in memory of Betty.
- 13. Contents Introduction page ix PART I UNIVERSAL CONSTRUCTIONS 1 1 The Stone-cech compactification βT 3 Appendix 7 2 Flows and their enveloping semigroups 19 3 Minimal sets and minimal right ideals 27 4 Fundamental notions 37 5 Quasi-factors and the circle operator 54 Appendix 60 PART II EQUIVALENCE RELATIONS AND AUTOMORPHISMS OF FLOWS 65 6 Quotient spaces and relative products 67 7 Icers on M and automorphisms of M 83 8 Regular flows 103 9 The quasi-relative product 111 PART III THE τ-TOPOLOGY 125 10 The τ-topology on Aut (X) 127 11 The derived group 138 vii
- 14. viii Contents 12 Quasi-factors and the τ-topology 149 PART IV SUBGROUPS OF G AND THE DYNAMICS OF MINIMAL FLOWS 155 13 The proximal relation and the group P 157 14 Distal flows and the group D 167 15 Equicontinuous flows and the group E 178 Appendix 193 16 The regionally proximal relation 203 PART V EXTENSIONS OF MINIMAL FLOWS 209 17 Open and highly proximal extensions 211 Appendix 222 18 Distal extensions of minimal flows 229 19 Almost periodic extensions of minimal flows 242 20 A tale of four theorems 257 References 264 Index 266
- 15. Introduction To a large extent this book is an updated version of Lectures on Topological Dynamics by Robert Ellis [Ellis, R., (1969)]. That book gave an exposition of what might be called an algebraic theory of minimal sets. Our goal here is to give a clear, self contained exposition of a new approach to the theory which allows for more straightforward proofs and develops a clearer language for expressing many of the fundamental ideas. We have included a treatment of many of the results in the aforementioned exposition, in addition to more recent developments in the theory; we have not attempted, however, to give a complete or exhaustive treatment of all the known results in the algebraic theory of minimal sets. Our hope is that the reader will be motivated to use the language and techniques to study related topics not touched on here. Some of these are mentioned either in the exercises or notes given at the end of various sections. This book should be suitable for a graduate course in topological dynamics whose prerequisites need only include some background in topology. We assume the reader is familiar with compact Hausdorff spaces, convergence of nets, etc., and perhaps has had some exposure to uniform structures and pseudo metrics which play a limited role in our exposition. A flow is a triple (X, T, π) where X is a compact Hausdorff space, T is a topological group, and π : X × T → X is a continuous action of T on X, so that xe = x and (xt)s = x(ts) for all x ∈ X, s, t ∈ T . Here we write xt = π(x, t) for all x ∈ X and t ∈ T , and e is the identity of the group T . Usually the symbol π will be omitted and the flow (X, T, π) denoted by (X, T ) or simply by X. In the situations considered here there is no loss of generality if T is given the discrete topology. The assumptions made thus far do not suffice to produce an interesting theory. The group T may be too “small” in its action on X. Thus for example, the trivial case where xt = x0, a fixed element of X, for all x ∈ X and t ∈ T , is not ruled out. To eliminate such degenerate behavior it is convenient to assume that the flow (X, T ) is point ix
- 16. x Introduction transitive, i.e. that there exists x0 ∈ X such that its orbit x0T ≡ {x0t | t ∈ T } is dense in X. The category P of point transitive flows has the remarkable property that it possesses a universal object; i,e, there exists a point transitive flow (βT, T ) such that any flow in P is a homomorphic image of βT . (See section 1 for a description of (βT, T ) and the proof of its universality.) Moreover, one may associate in a canonical fashion with any flow (X, T ) a point transitive flow E(X, T ). The latter, called the enveloping semigroup has proved extremely useful in the study of the dynamical properties of the original flow (X, T ). The enveloping semigroup is defined and studied in section 2, and examples of its use are scattered throughout the subsequent sections. This exposition focuses, however, on the category, M, of minimal flows. These are flows for which the orbit of every point x ∈ X is dense; that is xT = {xt | t ∈ T } = X for all x ∈ X. Again there exists a (unique up to isomorphism) universal object M in M. This fact was exploited in several papers to develop an “algebraic theory” of minimal flows. In particular a group was associated with each such flow and various relations among minimal flows studied by means of these groups. One purpose of this volume is to collect in one place the techniques which have proved useful in this study; another goal is to provide an exposition of a new approach to this material. The account of this algebraic theory of minimal flows given in Lectures on Topological Dynamics depends heavily on an algebraic point of view derived by studying the collection C(X) of continuous functions on X rather than X itself. In this volume we instead exploit the fact that X, as a homomorphic image of M, is of the form M/R for some icer (invariant closed equivalence relation) on M. We study the flow (X, T ) via the icer R rather than the algebra C(X). Another change is that the role of the group of automorphisms of a flow is emphasized. In particular the group G of automorphisms of M plays a crucial role. It is used both to codify the algebraic structure of M, and to define the groups associated to the minimal flows in M. In the earlier approach G was viewed as a subset Mu of M, where u ∈ M was a fixed idempotent. The new approach eliminates the asymmetrical treatment of the idempotents. Instead we view M = G(u) as a disjoint union (taken over all the idempotents in u ∈ M) of the images of the idempotents under the group G. Thus we explicitly take advantage of the fact that every p ∈ M can be written uniquely in the form α(u) with α ∈ G and u an idempotent in M. This approach also makes reliance on the concept of a pointed flow unnecessary. Previously the concept of a pointed flow was used to define, up to conjugacy, the group of a minimal flow; a different choice of base point corresponding to a conjugate
- 17. Introduction xi subgroup of Mu. From the point of view of icers on M, the group of the flow M/R is the subgroup: G(R) = {α ∈ G | gr(α) ⊂ R}, of G. Here gr(α) = {(p, α(p)) | p ∈ M} is the graph of the automorphism α of M. Again if S is an icer with M/R ∼ = M/S, then G(S) is conjugate to G(R). One of the important tools for the study of minimal flows is the so-called τ-topology on G. In section 10 we show how one can define a topology on the automorphism group Aut(X) of any regular flow (X, T ). Since G = Aut(M) and (M, T ) is regular, this allows one to define a topology on G. This topology on G coincides with the original definition of the τ-topology. (The idea for this viewpoint stems from J. Auslander’s approach to the τ-topology–private communication.) We would now like to make a few comments on some of the results which have been included herein. In part I we lay the foundation for what follows by treating the universal constructions upon which much of the later material is based. This includes an introduction to βT , the enveloping semigroup, and the universal minimal flow. The flow (2X, T ) whose minimal subflows are the so- called quasi-factors of the minimal flow (X, T ) is discussed in section 5. Here 2X is the collection of non-empty closed subsets of X. The space 2X is given the Vietoris topology detailed in the appendix to section 5. The extension of the action of T on 2X to an action of βT on 2X via the circle operator is also discussed in section 5, and used later in sections 12 and 17. Part II develops many of the techniques and language critical to our approach. As mentioned above, this approach hinges on identifying minimal flows as quotients of M by icers. We need not only to associate to any minimal flow an icer on M, but to any icer on M a minimal flow. The basic topological result needed is that the quotient of any compact Hausdorff space by a closed equiv- alence relation is again a compact Hausdorff space. Section 6 includes a proof of this result and a discussion of the relative product of two relations, a useful tool for constructing equivalence relations. The fundamental result concerning icers on M is proven in section 7. We show that any icer R on M can be written as a relative product R = (R ∩ P0) ◦ gr(G(R)) where P0 = {(α(u), α(v)) | α ∈ G and u, v are idempotents in M} (see 7.21). Regular flows, whose original definition is motivated in terms of automor- phisms, are those flows whose representation as a quotient M/R is unique. The flow (M, T ) is of course regular, and its structure serves as a prototype
- 18. xii Introduction for the algebraic structure of regular flows outlined in section 8. In particular, if (X, T ) is a regular flow, then the pair {X, Aut(X)} has properties analogous to those of the pair {M, G}, some of which were alluded to above. In part III we give a detailed exposition of the approach to the τ-topology mentioned earlier. When applied to the group Aut(X), for any regular minimal flow (X, T ), we obtain a topology which is compact and T1 but not Hausdorff. The construction of a derived group F for any closed subgroup F ⊂ Aut(X) is given in section 11. F is a normal subgroup of F which measures the extent to which F fails to be Hausdorff; in fact for any closed subgroup H ⊂ F, the quotient space F/H is Hausdorff if and only if F ⊂ H (see 11.10). In section 12 we give a proof of the fact that there exists a minimal flow X whose group G(X) = A if and only if A is a τ-closed subgroup of G. One example of such a flow is M/R where R = gr(A) = {gr(α) | α ∈ A}. The basic idea of the proof, which uses the material on quasi-factors, is the same as in Lectures on Topological Dynamics but the language of the current approach allows a more efficient treatment. Part IV is motivated by the questions: How are the various subgroups of G related to one another, and what do they tell us about the dynamics of minimal flows? It has long been known that the subcategories D and E of minimal distal and minimal equicontinuous flows respectively also possess universal objects XD and XE . Heretofore the groups D and E have been defined as the groups associated to these flows, i.e. D = G(XD) and E = G(XE ). In sections 14 and 15 we obtain intrinsic characterizations of D (see 14.6), and E (see 15.23) respectively. This gives content to the statements: if X is distal, then D ⊂ G(X), and if X is equicontinuous, then E ⊂ G(X). In fact, emphasizing the language of icers, M/R is distal (respectively equicontinuous) if and only if R = P0 ◦ gr(A) with D ⊂ A (respectively E ⊂ A). For proofs see 14.10 and 15.14 respec- tively. In particular distal and equicontinuous flows are completely determined by their groups. We show in 15.21 that G D = E, from which it follows immediately that (X, T ) is equicontinuous if and only if (X, T ) is distal and G ⊂ G(X). In section 13 we discuss the proximal relation P(X) on a min- imal flow X. In analogy with the distal and equicontinuous cases, we give a description of a subgroup P ⊂ G and show that P(X) is an equivalence rela- tion if and only if P ⊂ G(X). Another subgroup GJ ⊂ G is introduced and we show that P(X) is an equivalence relation with closed cells if and only
- 19. Introduction xiii if PGJ ⊂ G(X). In fact PGJ ⊂ = D which is consistent with the well-known result that P(X) is a closed invariant equivalence relation on X if and only if D ⊂ G(X). (see 14.8) In section 15 the regionally proximal relation, Q(X) of a minimal flow (X, T ) is introduced to facilitate the study of equicontinuous flows. (Recall that (X, T ) is equicontinuous if and only if Q(X) = X the diagonal in X × X.) The case Q ≡ Q(M) is also used to define the group E. Equicontinuous minimal flows are discussed from the point of view of icers on M in the body of section 15, while the approach to the same material via the enveloping semigroup is treated in the appendix. Q(X) is discussed in further detail in section 15 where we give a new proof of the fact that if E ⊂ G G(X), then Q(X) is an equivalence relation. To a large extent part V is concerned with generalizing the results of part IV to homomorphisms (extensions) of minimal flows. For instance for icers R ⊂ S on M, the canonical projection M/R → M/S is a distal homomorphism if and only if S = (R ∩ P0) ◦ gr(G(S)), moreover the extension is equicontinuous if and only if G(S) ⊂ G(R). We close with a section devoted to four theorems all of which are equivalent to the Furstenberg structure theorem for distal extensions; this section uses the language of icers and the techniques developed in the earlier sections to give proofs that all four theorems are equivalent. This fact does not seem to have been emphasized in the literature, and provides a good opportunity to illus- trate the language and techniques developed in the book. This analysis also illustrates an interesting twist to the icer approach. Here not only does the structure of the icers R and S come into play in understanding the extension M/R → M/S, the dynamics of the icer on M/R whose quotient gives M/S also plays an important role. The construction of the so-called Furstenberg tower provides another nice illustration of the language of icers; the stages in the tower are explicitly constructed using icers which are themselves con- structed from the groups involved. Section 20 itself does not contain the proof of the Furstenberg theorem. Instead we give a chart describing where proofs of various special cases appear in the text. On the other hand a complete proof for compact Hausdorff spaces, of the fact that any icer on a minimal flow which is both topologically tran- sitive and pointwise almost periodic must be trivial (one of the equivalents of the Furstenberg structure theorem) appears in 9.13. This is because our proof relies on the concept of the quasi-relative product developed in section 9. Indeed the quasi-relative product arose during our attempt to give a proof of
- 20. xiv Introduction the Furstenberg theorem in its full generality. The metric case of the theorem follows immediately from the fact that for metric flows the notions of point- transitivity and topological transitivity coincide. Our proof in the general case proceeds by reducing it to the metric case; the key tool in the construction which enables this is the quasi-relative product. While the quasi-relative prod- uct is only necessary for the most general version of the Furstenberg theorem, it turns out to be closely connected to quasi-factors (hence the name) and RIC extensions. We detail these connections in sections 9 and 17 respectively. A word about format We have written this book using a theorem-proof format. All the proofs are given using a sequence of numbered steps for which reasons are given at each stage. There are two main reasons for this approach. The first is to make sure that the arguments are as clear and accessible as possible. We found that insist- ing on numbering our steps and giving reasons forced a rigor, clarity, and atten- tion to detail we hope the reader will appreciate. We have attempted to avoid situations where as the material becomes more complex the reader is expected to fill in more gaps in the arguments. In addition to a better understanding of the details of the individual argu- ments, we hope that the format adds to the clarity of the overall exposition. The assumptions and conclusions of each of the lemmas, propositions, and theorems are stated carefully and precisely in a consistent format. These items are all numbered so that they can be referred to in a precise and unambiguous way as the exposition proceeds. We have tried to keep the proofs reasonably short and have divided the material into short sections, typically ten to fifteen pages long. In addition, we begin each section with an introduction designed to give an informal outline and motivation for the material in that section. The reader who wishes to go lightly on the intricate details, may wish to follow the train of thought by focusing on the introductions to each section and skipping the proofs. In this case, if a specific result attracts the reader’s interest, then the numbering system should facilitate a more careful reading of the details. This format is designed especially for the student who is not yet an expert; it assures that careful attention is paid to the details and that the train of thought is readily accessible.
- 21. PART I Universal constructions Our focus in the first part of this book is on the construction of certain universal objects that are crucial to the algebraic approach to the study of the asymptotic behavior of dynamical systems (flows). For the purposes of this exposition a flow is a pair (X, T ), where X is a compact Hausdorff space, and T is a group which acts on X (on the right). A homomorphism of flows is a continuous map- ping which preserves the actions. When the orbit closure of some point x0 ∈ X is all of X, that is x0T = X, we say that the flow (X, T ) is point transitive. If xT = X for all x ∈ X we say that (X, T ) is minimal. The collection of point transitive flows has a universal object, (βT, T ), in the sense that every point transitive flow is a homomorphic image of (βT, T ) (see 2.5). The action of T on βT extends in a natural way to a semigroup structure on βT which plays an important role in the study of flows. In section 1, we give an exposition of the structure of βT , relegating its construction via ultrafilters on T to an appendix. We exploit the properties of βT in section 2 to give a treatment of the enveloping semigroup E(X, T ) of a flow (X, T ). In Section 4 βT and E(X, T ) are used to introduce many of the fundamental notions which will be studied throughout the book. Of particular importance is the structure of the minimal ideals in E(X, T ) discussed in section 3. The fact that βT is its own enveloping semigroup allows us to apply these ideas to a minimal right ideal M ⊂ βT . On the other hand for such a minimal ideal, the flow (M, T ) is a universal object for the collection of all minimal flows (see 3.16). Our approach to the study of minimal flows involves exploiting the structure of M and the group of automorphisms of M to gain an understanding of the structure of the icers (closed invariant equivalence relations) on M. These ideas are pursued further in section 7 of Part II. Another construction which will play a significant role in our exposition is that of a quasi-factor of the flow (X, T ); this is by definition a subflow of the flow (2X, T ). Here by 2X we mean the space whose elements are closed non- empty subsets of X. In the appendix to section 5, we give an outline of the construction of the Vietoris topology, a compact Hausdorff topology on 2X. In the body of section 5 we develop some of the properties of the flow (2X, T ), including the extension of the natural action of T on 2X to an action of βT on 2X given by the so-called circle operator.
- 23. 1 The Stone-cech compactification βT The Stone-Cech compactification βT , is a compact Hausdorff space contain- ing the discrete group T as a dense subset. Of course one can construct the Stone-Cech compactification of any discrete set; a construction via ultrafilters is outlined in the appendix to this section. On the other hand βT is character- ized by certain properties which we take as its definition for the purposes of this section. When T is a group there is a natural semigroup structure on βT , for which left multiplication by all elements, and right multiplication by ele- ments of T are continuous. This semigroup structure plays a fundamental role in our study. In proposition 1.3 we deduce this structure as a consequence of the characterizing properties of βT ; in the appendix the semigroup structure is defined directly in terms of ultrafilters. Definition 1.1 Let T be a set with the discrete topology. The Stone-Cech compactification βT of T is determined up to homeomorphism by the fol- lowing properties: (i) T ⊂ βT with T = βT , (ii) βT is a compact Hausdorff space, and (iii) if X is a compact Hausdorff space and f :T → X, then there exists a unique continuous extension ˆ f : βT → X. The uniqueness of the extension in (iii) above is crucial. For instance it has as a consequence the fact that βT is unique up to homeomorphism. Indeed if Y is any space satisfying (i), (ii), and (iii), then the inclusions T ⊂ Y and T ⊂ βT extend to continuous maps ϕ : βT → Y and ψ : Y → βT . The composition ϕ ◦ ψ is thus a continuous extension of the inclusion T ⊂ Y to Y, and hence by uniqueness must be the identity. Similarly ψ ◦ ϕ is the identity on βT , and therefore ϕ is a homeomorphism with inverse ψ. This shows that as a topological space βT is completely determined by the conditions in 1.1. 3
- 24. 4 The Stone-cech compactification βT The following theorem confirms this by exhibiting a base for the topology on βT . It is interesting to note that this base consists of sets which are both open and closed in βT . Theorem 1.2 Let: (i) T be a set with the discrete topology and βT be its Stone-Cech compact- ification, (ii) A ⊂ T , and (iii) V ⊂ βT be an open set. Then: (a) βT = A ∪ T A is a disjoint union, and thus A is both open and closed (clopen) in βT , (b) V = V ∩ T , and hence V is both open and closed, and (c) {A | A ⊂ T } is a base for the topology on βT . PROOF: (a) 1. Let ∅ = A ⊂ T . 2. Let χA : T → {0, 1} be defined by χA(t) = 1 if t ∈ A 0 otherwise . 3. There exists a continuous extension χ̂A : βT → {0, 1}. (by 2, 1.1(iii)) 4. χ̂−1 A (1) and χ̂−1 A (0) are clopen with A ⊂ χ̂−1 A (1) and T A ⊂ χ̂−1 A (0). (by 2, 3) 5. Let p ∈ χ̂−1 A (1) and W ⊂ βT be open with p ∈ W. 6. There exists t ∈ T with t ∈ W ∩ χ̂−1 A (1). (by 4, 5, 1.1(i)) 7. t ∈ A ∩ W. (by 2, 3, 6) 8. p ∈ A. (by 5, 7) 9. χ̂−1 A (1) ⊂ A. (by 5, 8) 10. χ̂−1 A (0) ⊂ T A. (similar argument) 11. χ̂−1 A (1) = A and χ̂−1 A (0) = T A. (by 4, 9, 10) (b) 1. Clearly V ∩ T ⊂ V . 2. Let W ⊂ βT be open and p ∈ V ∩ W. 3. There exists t ∈ T with t ∈ V ∩ W. (by 2, 1.1(i)) 4. t ∈ (V ∩ T ) ∩ W. (by 3) 5. p ∈ V ∩ T . (by 2, 4) (c) 1. Let ∅ = V ⊂ βT be open and p ∈ V . 2. There exists W open with p ∈ W ⊂ W ⊂ V . (βT is compact Hausdorff) 3. p ∈ W = W ∩ T ⊂ V . (by 2, part (b)) 4. {A | A ⊂ T } is a base for the topology on βT . (by 1, 3) We will be most interested in the space βT when T is a group. In this case, and in fact whenever T is a semigroup, the semigroup structure on T induces a
- 25. The Stone-cech compactification βT 5 semigroup structure on βT . Once again the uniqueness of the extension in 1.1 (iii) is crucial. The following proposition details the construction. Proposition 1.3 Let T be a semigroup, so that T is provided with an associa- tive binary operation: T × T → T (s, t) → st. Then the semigroup structure on T extends to one on βT , βT × βT → βT (p, q) → pq such that: (a) the right multiplication map Rt : βT → βT p → pt is continuous for all t ∈ T , and (b) the left multiplication map Lp : βT → βT q → pq is continuous for all p ∈ βT . PROOF: 1. Let mt (s) = st for all s, t ∈ T . 2. There exists a continuous extension Rt : βT → βT of mt for every t ∈ T . (by (iii) of 1.1) 3. There exists a continuous extension Lp : βT → βT of the map T → βT t → Rt (p) . (by (iii) of 1.1) 4. For p, q ∈ βT we define pq ≡ Lp(q). 5. Let t, s ∈ T . Then the maps βT → βT and βT → βT p → (ps)t p → p(st) are both continuous extensions of the map T → βT t → (t s)t = t (st). 6. p(st) = (ps)t for all p ∈ βT and s, t ∈ T . (by 5 and uniqueness in (iii) of 1.1) 7. Let p, q ∈ βT . Then the maps βT → βT and βT → βT q → (pq)t q → p(qt) are both continuous extensions of the map T → βT s → (ps)t = p(st). (by 3, 6)
- 26. 6 The Stone-cech compactification βT 8. p(qt) = (pq)t for all p ∈ βT and t ∈ T . (by 7 and uniqueness in (iii) of 1.1) 9. The maps βT → βT and βT → βT r → (pq)r r → p(qr) are both continuous extensions of the map T → βT t → (pq)t = p(qt). ( by 3, 8) 10. p(qr) = (pq)r for all p, q, r ∈ βT . (by 9 and uniqueness in (iii) of 1.1) The space βT can be provided with a (different) semigroup structure in which left multiplication is continuous for all t ∈ T , and right multiplication is continuous for all p ∈ βT . Merely mimic the proof of 1.2 starting with the map mt : T → T s → ts . We will most often be interested in right actions of a group T . Henceforth we will always assume unless explicitly indicated otherwise that T is a group, and that βT is provided with the semigroup structure of 1.2. In the upcoming sections we will make extensive use of this semigroup structure and in particular the fact that it makes (βT, T ) into a flow. It is important to note that the assumption that T is a group, so that every element of T has an inverse does not guarantee that the elements of βT have inverses. In fact βT is a group only if T is finite and βT = T . In general, the only elements of βT which have inverses are the elements of T . This follows immediately from the fact that p, q ∈ βT with pq ∈ T implies that p, q ∈ T . Indeed if pq = t ∈ T , then t ∈ Lp(βT ) = Lp(T ) = Lp(T ) since Lp is continuous. On the other hand T ⊂ βT has the discrete topology so {t} is an open subset of βT . It follows that t ∈ Lp(T ) and there exists s ∈ T with ps = t. But this implies that p = ts−1 ∈ T and q = s ∈ T . We end this section with an elementary proposition which speaks to the naturality of the construction of βT . Proposition 1.4 Let: (i) T be a semigroup, (ii) ∅ = H ⊂ T , and (iii) j : βH → βT be the continuous extension to βH of the inclusion H → βT . Then: (a) j is injective, (b) im j = H, and
- 27. The Stone-cech compactification βT 7 (c) if H is a subsemigroup of T , then j(pq) = j(p)j(q) for all p, q ∈ βH. (Thus we will identify βH with H ⊂ βT .) PROOF: (a) 1. Let h0 ∈ H. 2. Let ϕ : T → βH be defined by ϕ(t) = t if t ∈ H h0 if t ∈ H . 3. Let ϕ̂ : βT → βH be the continuous extension of ϕ to βT . 4. Let ψ = ϕ̂ ◦ j : βH → βH. 5. ψ(h) = h for all h ∈ H. (by 2, 3, 4) 6. ψ(p) = p for all p ∈ βH. (by 3, (iii), 1.1(iii)) 7. j is injective. (by 4, 6) (b) and (c) We leave these to the reader. APPENDIX TO SECTION 1: ULTRAFILTERS AND THE CONSTRUCTION OF βT Our goal here is the construction of the compact Hausdorff space βT , which is characterized up to homeomorphism by 1.1. Those readers already familiar with ultrafilters will recall that a topological space X is compact if and only if every ultrafilter on X converges to a point in X (see 1.A.16 and Ex. 1.5). This motivates the approach we will take; in analogy with the construction of the real numbers as Cauchy sequences of rational numbers, βT will be iden- tified as the collection of ultrafilters on T . We have attempted to make this presentation self-contained, so that filters and ultrafilters are defined, and the elementary properties necessary for the construction explicitly introduced. We make use of one of these properties, namely 1.A.8, in the appendix to section 5. All the other sections of the book, while occasionally using the terminology of this appendix, rely only on the results of section 1 itself. In the interest of brevity, proofs of some of the results in this appendix are left as exercises for the reader. We begin with some background material on filters and ultrafilters. Definition 1.A.1 Let T be a nonempty set and F a collection of nonempty subsets of T . We make the following definitions: (a) F is a filter base on T if F1, . . . , Fn ∈ F =⇒ there exists F ∈ F with F ⊂ F1 ∩ · · · ∩ Fn. (b) Fc = {A | A ⊂ T and there exists F ∈ F with F ⊂ A}. (c) F is a filter on T if F is a filter base on T and Fc = F. Thus if F is a filter, then it has the finite intersection property (F.I.P.), meaning F1, . . . , Fn ∈ F =⇒ F1 ∩ · · · ∩ Fn ∈ F.
- 28. 8 The Stone-cech compactification βT (d) U is an ultrafilter on T if U is a filter on T such that F a filter on T with U ⊂ F =⇒ U = F (so that U is a maximal filter on T ). The neighborhoods of a point x in a topological space provide an important motivating example; we leave it as an exercise for the reader to verify this. Example 1.A.2 Let X be a topological space and x ∈ X. Then the collection Nx = {A | there exists U open in X with x ∈ U ⊂ A} is a filter on X. We refer to Nx as the neighborhood filter at x. Another elementary example which plays a fundamental role in the con- struction of βT is the following: Example 1.A.3 Let t ∈ T . Then the collection h(t) = {A | t ∈ A ⊂ T } is an ultrafilter on T . Moreover h(t) is the only ultrafilter on T which contains the singleton set {t}. We refer to h(t) as the principal ultrafilter generated by t. PROOF: We leave the proof as an exercise for the reader. According to 1.A.3, every t ∈ T generates an ultrafilter on T ; we now observe that any filter is contained in some ultrafilter. Suppose that {Fi | i ∈ I} is a collection of filters on T , where I is a totally ordered set. Assume further that if i j ∈ I, then Fi ⊂ Fj . (These assumptions amount to saying that this collection is an increasing chain of filters on T .) Then it is straightforward to check that the union i∈I Fi is a filter on T . This shows that every increasing chain of filters has a maximal element; hence as an immediate consequence of Zorn’s lemma (see also 3.3 for a statement) every filter is contained in some maximal filter (i.e an ultrafilter). We state this result as a lemma for future reference: Lemma 1.A.4 Let F be a filter (or filter base) on T . Then there exists an ultrafilter U on T such that F ⊂ U. The next few results examine the structure of ultrafilters on T . In particular they allow us to characterize those filters which are ultrafilters. In fact a filter F is an ultrafilter if and only if for every ∅ = A ⊂ T , either A or its complement lie in F. (This is the content of 1.A.6 and 1.A.7.)
- 29. The Stone-cech compactification βT 9 Proposition 1.A.5 Let: (i) U be an ultrafilter on T , and (ii) A ⊂ T . Then A ∈ U if and only if A ∩ U = ∅ for all U ∈ U. PROOF: 1. Since U is a filter it is clear that A ∈ U ⇒ A ∩ U = ∅ for all U ∈ U. 2. Assume that A ∩ U = ∅ for all U ∈ U. 3. Let G = {G ⊂ T | A ∩ U ⊂ G for some U ∈ U}. 4. Let G1, . . . , Gn ∈ G. 5. There exist Ui ∈ U such that A ∩ Ui ⊂ Gi for 1 ≤ i ≤ n. (by 3, 4) 6. U = U1 ∩ · · · ∩ Un ∈ U. (by 5, (i)) 7. A ∩ U ⊂ G1 ∩ · · · ∩ Gn. (by 5, 6) 8. G1 ∩ · · · ∩ Gn ∈ G. (3, 7) 9. G is a filter on T . (by 4, 8) 10. U ⊂ G. (by 3) 11. U = G. (by 10, (i)) 12. A ∈ U. (by 3, 11) Corollary 1.A.6 Let: (i) U be an ultrafilter on T , and (ii) A ⊂ T . Then either A ∈ U or T A ∈ U. PROOF: 1. Assume that A / ∈ U. 2. There exists U ∈ U such that A ∩ U = ∅. (by 1, 1.A.5) 3. U ⊂ T A. (by 2) 4. T A ∈ U. (by 3, (i)) Proposition 1.A.7 Let: (i) F be a filter on T , and (ii) A ∈ F or T A ∈ F for all A ⊂ T . Then F is an ultrafiter on T . PROOF: 1. Let G be a filter on T with F ⊂ G. 2. Let G ∈ G. 3. T G / ∈ G. (by 1, 2) 4. T G / ∈ F. (by 1, 3) 5. G ∈ F. (by 4, (ii))
- 30. 10 The Stone-cech compactification βT 6. G ⊂ F. (by 2, 5) 7. F is an ultrafilter on T . (by (i), 1, 5) The following natural generalization of 1.A.6 will be useful here and is used in proposition 5.A.3 of the appendix to section 5. Corollary 1.A.8 Let: (i) U be an ultrafilter on T , (ii) A1, . . . , An be subsets of T , and (iii) A1 ∪ · · · ∪ An ∈ U. Then there exists j with Aj ∈ U. PROOF: 1. Assume that Ai / ∈ U for all i = j. 2. T Ai ∈ U for all i = j. (by 1, (i), 1.A.6) 3. i=j (T Ai) ∈ U. (by 2, (i)) 4. Aj ∩ i=j (T Ai) = (A1 ∪ · · · ∪ An) ∩ i=j (T Ai) ∈ U. (by 3, (i), (iii)) 5. Aj ∈ U. (by 4, (i)) 6. There exists j with Aj ∈ U. (by 1, 5) Having discussed a few of the elementary properties of ultrafilters, we are ready to define the Stone-Cech compactification βT of T . As a set βT simply consists of all the ultrafilters on T ; the next step is to define a topology on βT . Describing this topology requires some notation. Definition 1.A.9 Let T be a nonempty set. We define βT by βT = {U | U is an ultrafilter on T }. Definition and Notation 1.A.10 Let ∅ = A ⊂ T . We define the hull of A by h(A) = {u ∈ βT | A ∈ u}. Note that for t ∈ T we have used the notation h(t) for the single element h(t) = {A | t ∈ A ⊂ T } ∈ βT, whereas the hull h({t}) as defined above is a subset of βT . This notation is justified by the fact that h({t}) = {h(t)} since h(t) is the only ultrafilter which contains {t}. We will identify T with the subset {h(t) | t ∈ T } ⊂ βT and thus write t for the element h(t) ∈ βT . Note that if an ultrafilter u ∈ h(A) ∩ h(B), then A ∈ u and B ∈ u. This implies that A ∩ B ∈ u and hence u ∈ h(A ∩ B). It follows that the collection {h(A) | A ⊂ T } is a base for a topology on βT . This gives us the following proposition.
- 31. The Stone-cech compactification βT 11 Proposition 1.A.11 Let T = { ⊂ βT | for every u ∈ there exists A ∈ u with h(A) ⊂ }. Then T is a topology on βT . Henceforth we will assume that βT is provided with the topology T . In this topology if A ⊂ T ⊂ βT , then h(A) is the closure of A in βT . We leave the proof as an exercise for the reader; one immediate consequence is the fact that A is both open and closed in βT , as we saw in 1.2. On the other hand every ultrafilter on T contains the set T , so βT = h(T ) = T , in other words T is dense in βT . We restate this for emphasis. Proposition 1.A.12 T is dense in βT . Having constructed the space βT satisfying condition (i) of 1.1, we now wish to show that βT is a compact Hausdorff space (condition (ii) of 1.1). Lemma 1.A.13 Let: (i) {Ai | i ∈ I} be nonempty subsets of T , and (ii) i∈I h(Ai) = βT . Then there exists a finite subset F ⊂ I with i∈F Ai = T . PROOF: 1. Assume that BF = T i∈F Ai = ∅ for all finite sets F ⊂ I. 2. Let B = {BF | F ⊂ I finite}. 3. BF1∪···∪Fn = T ⎛ ⎝ i∈F1∪···∪Fn Ai ⎞ ⎠ = n j=1 ⎛ ⎝T ⎛ ⎝ i∈Fj Ai ⎞ ⎠ ⎞ ⎠ = BF1 ∩ · · · ∩ BFn . (by 1) 4. B is a filter base on T . (by 2, 3) 5. There exists an ultrafilter U on T with B ⊂ U. (by 4, 1.A.4) 6. There exists k ∈ I with U ∈ h(Ak), and hence Ak ∈ U. (by 5, (ii)) 7. T Ak = B{k} ∈ B ⊂ U. (by 1, 2, 4) 8. BF = ∅ and hence i∈F Ai = T for some finite set F ⊂ T . (6, 7 contradict 1) Theorem 1.A.14 (βT, T ) is a compact Hausdorff topological space. PROOF: 1. Let { i | i ∈ I} be a family of open subsets of βT with i∈I i = βT . 2. For every u ∈ βT , there exists Au ∈ u and iu ∈ I with h(Au) ⊂ iu. (by 1) 3. u∈βT h(Au) = βT . (by 2) 4. There exists a finite subset F ⊂ βT . such that u∈F Au = T . (by 3, 1.A.13)
- 32. 12 The Stone-cech compactification βT 5. Let v ∈ βT . 6. Au ∈ v for some u ∈ F. (by 4, 5, 1.A.8) 7. v ∈ h(Au) ⊂ iu . (by 2, 6) 8. u∈F iu = βT . (by 5, 7) 9. (βT, T ) is compact. (by 1, 8) 10. Let u1 = u2 ∈ βT . 11. There exists a subset A ⊂ T such that A ∈ u1 and A / ∈ u2. (by 1) 12. T A ∈ u2. (by 11, 1.A.7) 13. u1 ∈ h(A) and u2 ∈ h(T A). (by 11, 12) 14. h(A) and h(T A) are disjoint open subsets of βT . 15. (βT, T ) is Hausdorff. (by 10, 13, 14) In order to show that βT satisfies the final condition of 1.1, we make use of the fact that in a compact Hausdorff space X, every ultrafilter converges to a unique point in X. This is the content of 1.A.16 whose proof we include in the interest of completeness. The converse also holds, but since we will not make explicit use of it, we have left its proof as an exercise. Definition 1.A.15 Let X be a topological space, x ∈ X and F be a filter on X. We say that F converges to x and write F → x, if the neighborhood filter at x, Nx ⊂ F. Lemma 1.A.16 Let: (i) X be a compact Hausdorff topological space, (ii) ∅ = Y ⊂ X, (iii) U be an ultrafilter on X, and (iv) Y ∈ U. Then there exists a unique x ∈ Y such that U → x. PROOF: 1. Assume that U → x for all x ∈ Y. 2. For every x ∈ Y there exists Vx ∈ Nx with Vx / ∈ U. (by 1) 3. There exists a finite subset F ⊂ Y such that Y ⊂ x∈F Vx. (by 2, (Y is compact by (i))) 4. Y = x∈F (Vx ∩ Y) ∈ U. (by 3, (iv)) 5. There exists x ∈ F such that Vx ∩ Y ∈ U. (by 4, (iii), 1.A.8) 6. There exists x ∈ Y such that U → x. (5 contradicts 2) 7. Uniqueness follows from the fact that X is Hausdorff. We now wish to prove that every mapping from T to a compact Hausdorff space Y can be extended to a continuous map βT → Y. The proof uses the fol- lowing elementary lemma, whose proof we leave as an exercise for the reader.
- 33. The Stone-cech compactification βT 13 Lemma 1.A.17 Let: (i) f : T → Y, (ii) U be an ultrafilter on T , and (iii) ¯ f (U) = {B ⊂ Y | there exists A ∈ U with f (A) ⊂ B}. Then ¯ f (U) is an ultrafilter on Y. Theorem 1.A.18 (Compare with 1.1.) Let: (i) X be a compact Hausdorff topological space, and (ii) f : T → X. Then there exists a unique continuous map ˆ f : βT → X with ˆ f (t) = f (t) for all t ∈ T . PROOF: 1. Let u ∈ βT . 2. ¯ f (u) = {B ⊂ X | there exists A ∈ u with f (A) ⊂ B} is an ultrafilter on X. (by 1, 1.A.17) 3. There exists a unique element ˆ f (u) ∈ X such that ¯ f (u) → ˆ f (u). (by 2, 1.A.16) 4. Let t ≡ h(t) ∈ T ⊂ βT . 5. ¯ f (t) = {B ⊂ X | f (t) ∈ B} ⊃ Nf (t). (by 2, 4) 6. ˆ f (t) = f (t). (by 3, 5) 7. Let V ⊂ X be open and u ∈ βT with u ∈ ˆ f −1(V ). 8. There exists W ⊂ X open with ˆ f (u) ∈ W ⊂ W ⊂ V . (by 7, (i)) 9. ¯ f (u) → ˆ f (u) ∈ W. (by 3, 8, 1.A.16) 10. W ∈ ¯ f (u). (by 8, 9) 11. There exists B ∈ u with f (B) ⊂ W. (by 10) 12. Let v ∈ h(B) (so that B ∈ v). 13. W ∈ ¯ f (v). (by 11, 12) 14. There exists x ∈ W such that ¯ f (v) → x. (by 13, 1.A.16) 15. ˆ f (v) = x ∈ V . (by 3, 8, 14) 16. v ∈ ˆ f −1(V ). (by 15) 17. u ∈ h(B) ⊂ ˆ f −1(V ). (by 11, 12, 16) 18. ˆ f −1(V ) is open and hence ˆ f is continuous. (by 7, 17) 19. ˆ f is unique because T is dense in βT (by 1.A.12) and X is Hausdorff. We turn now to the case where T is a semigroup. We saw in 1.3 that in this case there is a unique semigroup structure on βT which makes right multiplication by elements of T and left multiplication by all elements of βT continuous. Having constructed the topological space βT using the ultrafilters on T , we complete this appendix by showing that a semigroup structure on T allows us to define the product of two ultrafilters on T , giving βT a semigroup structure.
- 34. 14 The Stone-cech compactification βT This definition is motivated by the fact that for an ultrafilter u ∈ βT , and an element t ∈ T , one expects their product to be given by: ut = {At | A ∈ u} = {A | At−1 ∈ u} which in fact makes sense when T is a group but may not be an ultrafilter when the right multiplication map Rt is not one-one. Instead we will use ut = {A | R−1 t (A) ∈ u}, which gives the same result when T is a group since R−1 t = Rt−1 in that case. Thus the ultrafilter ut = uh(t) should be characterized by the fact that: A ∈ ut ⇐⇒ t ∈ {s | R−1 s (A) ∈ u} ⇐⇒ {s | R−1 s (A) ∈ u} ∈ h(t). This viewpoint generalizes in a natural way to the product uv of two ultrafil- ters: A ∈ uv ⇐⇒ {s | R−1 s (A) ∈ u} ∈ v. There are many details to check; for instance it needs to be shown that uv as defined above is an ultrafilter. We content ourselves with giving a precise outline of the notation and results involved while leaving the details of the proofs to the reader. Definition and Notation 1.A.19 Let T be a semigroup, t ∈ T , A ⊂ T , and u ∈ βT . We use the notation: Rt : T → T Lt : T → T s → st s → ts for the multiplication maps in T , and At = Rt (A) = {at | a ∈ A}. We define A ∗ u = {s ∈ T | R−1 s (A) ∈ u}. Note that R−1 s (A) ∈ h(t) ⇐⇒ t ∈ R−1 s (A) ⇐⇒ ts ∈ A ⇐⇒ s ∈ L−1 t (A), so that A ∗ h(t) = L−1 t (A).
- 35. The Stone-cech compactification βT 15 Proposition 1.A.20 Let: (i) T be a semigroup, (ii) u, v ∈ βT , and (iii) w = {A ⊂ T | A ∗ u ∈ v}. Then w is an ultrafilter on T . PROOF: We leave the proof as an exercise for the reader. Definition 1.A.21 Let T be a semigroup, and u, v ∈ βT . We define uv = {A ⊂ T | A ∗ u ∈ v}, so that uv ∈ βT by 1.A.20. For any two elements s, t ∈ T , applying 1.A.21 yields: h(s)h(t) = {A | A ∗ h(s) ∈ h(t)} = {A | t ∈ L−1 s (A)} = {A | st ∈ A} = h(st), so this definition of a product on βT agrees with the semigroup structure on T . It follows immediately from the next lemma that this product is associative and hence gives βT a semigroup structure which extends the semigroup structure on T . Lemma 1.A.22 Let: (i) T be a semigroup, (ii) u, v ∈ βT , and (iii) ∅ = A ⊂ T . Then A ∗ (uv) = (A ∗ u) ∗ v. PROOF: We leave the proof as an exercise for the reader. Proposition 1.A.23 Let: (i) T be a semigroup, and (ii) u, v, w ∈ βT . Then (uv)w = u(vw). PROOF: A ∈ (uv)w ⇐⇒ (A ∗ u) ∗ v = A ∗ (uv) ∈ w ⇐⇒ A ∗ u ∈ vw ⇐⇒ A ∈ u(vw). (by 1.A.22) We complete our discussion of βT by indicating how to verify that the maps Rt and Lp are continuous for all t ∈ T and p ∈ βT respectively. Lemma 1.A.24 Let: (i) T be a semigroup, (ii) ∅ = A ⊂ T ,
- 36. 16 The Stone-cech compactification βT (iii) t ∈ T , (iv) u ∈ βT , (v) Rt : βT → βT v → vt , and (vi) Lu : βT → βT. v → uv Then (a) R−1 t (h(A)) = h(R−1 t (A)), (b) L−1 u (h(A)) = h(A ∗ u). PROOF: We leave the proof as an exercise for the reader. Proposition 1.A.25 (Compare with 1.3) Let: (i) T be a semigroup, (ii) t ∈ T , (iii) u ∈ βT , (iv) Rt : βT → βT, v → vt and (v) Lu : βT → βT v → uv . Then Rt and Lu are both continuous. PROOF: This follows immediately from 1.A.24. EXERCISES FOR CHAPTER 1 Exercise 1.1 (See 1.4) Let: (i) T be a semigroup, (ii) ∅ = H ⊂ T be a subsemigroup, and (iii) j : βH → βT be the continuous extension to βH of the inclusion H → βT . Then j(pq) = j(p)j(q) for all p, q ∈ βH (so that βH ≡ H is a subsemi- group of βT ). Exercise 1.2 Let X be a topological space and x ∈ X. Prove that the collection Nx = {A | there exists U open in X with x ∈ U ⊂ A} is a filter on T .
- 37. The Stone-cech compactification βT 17 Exercise 1.3 Let: (i) F be a filter on T , and (ii) U(F) = {U | U is an ultrafilter on T with F ⊂ U}. Prove that F = U∈U(F) U. Exercise 1.4 Let t ∈ T . Prove that the collection h(t) = {A | t ∈ A ⊂ T } is the unique ultrafilter on T containing {t}. Exercise 1.5 Let X be a topological space. Show that X is compact if and only if every ultrafilter on X converges. (Compare with 1.A.16.) Exercise 1.6 Let: (i) f : T → Y, (ii) U be an ultrafilter on T , and (iii) ¯ f (U) = {B ⊂ Y | there exists A ∈ U with f (A) ⊂ B}. Show that ¯ f (U) is an ultrafilter on Y. Exercise 1.7 Let T = { ⊂ βT | for every u ∈ there exists A ∈ u with h(A) ⊂ }. Show that T is a topology on βT ; since ∅, βT ∈ T this amounts to proving: (a) if { i | i ∈ I} ⊂ T , then i∈I i ∈ T , and (b) if { 1, . . . , n} ⊂ T , then 1 ∩ · · · ∩ n ∈ T . Exercise 1.8 Let: (i) T = { ⊂ βT | ∀ u ∈ , ∃A ∈ u with h(A) ⊂ }, and (ii) A ⊂ T . Show that A = h(A); so that A is both open and closed with respect to the topology T . Exercise 1.9 Let: (i) T be a semigroup, (ii) u, v ∈ βT , and (iii) uv = {A ⊂ T | A ∗ u ∈ v}.
- 38. 18 The Stone-cech compactification βT Show that (a) uv is an ultrafilter on T , and (b) A ∗ (uv) = (A ∗ u) ∗ v for all ∅ = A ⊂ T . Exercise 1.10 Let: (i) T be a semigroup, (ii) ∅ = A ⊂ T , (iii) t ∈ T , (iv) u ∈ βT , (v) Rt : βT → βT, v → vt and (vi) Lu : βT → βT, v → uv Show that (a) R−1 t (h(A)) = h(R−1 t (A)), and (b) L−1 u (h(A)) = h(A ∗ u).
- 39. 2 Flows and their enveloping semigroups For us a flow will be a compact Hausdorff space X provided with a continuous (right) action of a group T on X. In Topological Dynamics we are concerned with the so-called asymptotic behavior of this action. This motivates the con- sideration of not just the collection T thought of as a subset of the set XX of self-mappings of X, but all of the limit points of T in XX. This gives rise to the notion of the enveloping semigroup E(X, T ) of the flow. The compo- sition of functions gives a natural semigroup structure on E(X, T ) which as we will see in subsequent sections, can be exploited to study the asymptotics of the original flow (X, T ). The semigroup structure on βT discussed in the previous section and its appendix serves as a prototype example. Indeed since T ⊂ βT , this semigroup structure makes (βT, T ) itself a flow, and we will see in proposition 2.9 that E(βT, T ) ∼ = βT in a natural way. In this section we introduce the appropriate notation, the details of the con- struction, and give an exposition of some of the elementary properties of E(X, T ). Many of these properties will be used directly, and serve as moti- vation in what follows. Several accounts of this material appear in the text- book-literature (see for example [Auslander, J., (1988)] and [Ellis, R., (1969)]. However, the point of view we have adopted here is slightly different. In order to emphasize the connection between the two, we have involved βT in the definition of E(X, T ) (see 2.8). In particular the fact that E(X, T ) is a homo- morphic image of βT , both as a flow and as a semigroup has important conse- quences. We begin with some basic notation and definitions. Definition 2.1 Let X be a set and S a semigroup. Then an action of S on X is a function π : X × S → X (x, s) → xs such that x(st) = (xs)t 19
- 40. 20 Flows and their enveloping semigroups for all x ∈ X and s, t ∈ S. If S has an identity e we require that xe = x for all x ∈ X. If π is an action of S on X, we say that S acts on X via π or simply that S acts on X. Definition 2.2 A flow is a triple (X, T, π) where X is a compact Hausdorff space, T is a topological group, and π is an action of T on X such that the map π is continuous. Let A ⊂ X. We say that A is invariant if AT ≡ {at | a ∈ A, t ∈ T } ⊂ A. If A is invariant, then the restriction of π to A × T defines an action of T on A. If A is also closed, the resulting flow (A, T, π) is called a subflow of the flow (X, T, π). Most of the time the symbol π is suppressed, i.e. the flow (X, T, π) is denoted (X, T ) or simply, X. For all of the flows (X, T, π) considered here we will assume that T is pro- vided with the discrete topology. In this case for π to be continuous it suffices that the maps πt : X → X x → xt be continuous for all t ∈ T . As we mentioned above, it follows from 1.3 that the map π : βT ×T → βT defined by π(p, t) = pt makes (βT, T ) a flow. We now make explicit the definition of a homomorphism of flows mentioned earlier. Definition 2.3 Let (X, T ), (Y, T ) be flows and f : X → Y. Then f is a homo- morphism if f is continuous and f (xt) = f (x)t for all x ∈ X, and t ∈ T . The set of homomorphisms from X to Y will be denoted Hom(X, Y). The set of automorphisms (bijective elements of Hom(X, X)) of X, will be denoted Aut(X). One example of an asymptotic property of a flow is point transitivity as defined below. Definition 2.4 Let (X, T ) be a flow. We say that (X, T ) is point transitive if there exists x0 ∈ X with x0T = X. Clearly the flow (βT, T ) is point transitive since eT = T = βT . Note also that if f : (X, T ) → (Y, T ) is an epimorphism (surjective homomorphism), and (X, T ) is point transitive, then (Y, T ) is point transitive. Thus any homo- morphic image of (βT, T ) is point transitive. As we show in 2.5, the converse of this statement also holds. For this reason the flow (βT, T ) is often called the universal point transitive flow. Proposition 2.5 Let (X, T ) be a point transitive flow. Then there exists an epimorphism f : (βT, T ) → (X, T ).
- 41. Flows and their enveloping semigroups 21 PROOF: 1. There exists x0 ∈ X with x0T = X. 2. The map T → X t → x0t has a unique continuous extension f : βT → X p → x0p . (by 1.1) 3. Let t ∈ T . 4. The maps βT → X p → x0(pt) and βT → X p → (x0p)t are continuous extensions of the map T → X t → (x0t )t = x0(t t). 5. x0(pt) = (x0p)t and hence f is a homomorphism. (by 2, 3, 4, 1.1) 6. x0T = f (T ) ⊂ f (βT ). (by 1) 7. X = x0T ⊂ f (βT ). (by 2, 6, since βT and X are compact Hausdorff spaces) In order to describe the so-called enveloping semigroup of a flow (X, T ) we introduce the space of self-maps of X, along with some useful notation. Since our actions are on the right, it is natural write xπt = xt for the value at x of the element πt of XX associated with t ∈ T . For this reason we will write all of the functions in XX on the right. Notation 2.6 Let X be a compact Hausdorff space. Then XX will denote the set of maps of X to X provided with the topology of pointwise convergence. Let f, g ∈ XX and x ∈ X. Then xf will denote the image of x under f , and fg the composite map first f then g. Thus x(fg) = (xf )g. Finally ρ : XX × XX → XX will denote the map defined by ρ(f, g) = fg for all f, g ∈ XX. We will make use of the following elementary properties of XX. Proposition 2.7 Let X be compact Hausdorff. Then: (a) XX is compact Hausdorff, (b) ρ provides XX with a semigroup structure, (c) the maps ρf : XX → XX g → gf are continuous for all continuous f ∈ XX, (d) the maps ρf : XX → XX g → fg are continuous for all f ∈ XX, and (e) ρ defines an action of the semigroup XX on the set XX.
- 42. 22 Flows and their enveloping semigroups PROOF: (a) and (b) are standard. (c) 1. Let gα −→ g be a convergent net, and f be continuous. 2. xgα −→ xg for all x ∈ X. 3. x(gαf ) = (xgα)f −→ 1,2 (xg)f = x(gf ) for all x ∈ X. 4. gαf −→ gf . (by 3) (d) 1. Let gα −→ g. 2. ygα −→ yg for all y ∈ X. (by 1) 3. x(fgα) = (xf )gα−→ 2 (xf )g = x(fg) for all x ∈ X. 4. fgα −→ fg. (by 3) (e) This follows immediately from parts (b) and (c). It is clear that the map X × XX → X (x, f ) → xf defines an action of the semigroup XX on X. Thus if T is any subgroup of XX which consists entirely of continuous maps, then (X, T ) is a flow. Conversely given any flow (X, T ), the set {πt | t ∈ T } is a subgroup of XX consisting of continuous maps. In this case we obtain a group homomorphism of T into a subgroup of XX, allowing the following definition of the enveloping semigroup of the flow (X, T ). Definition 2.8 Let (X, T ) be a flow. By 1.1 the map T → XX t → πt has a con- tinuous extension X : βT → XX. The image of X, X(βT ) = {πt | t ∈ T } ≡ E(X, T ) is clearly a subsemigroup of XX, which we call the enveloping semigroup of the flow (X, T ) and denote by E(X, T ) or simply E(X). The map X will be referred to as the canonical map of βT onto E(X). Since we write xt for xπt = x X(t) for all x ∈ X and t ∈ T , it will be convenient to write xp for x X(p) for all x ∈ X and p ∈ βT . Some of the properties of E(X) which follow directly from the definition are collected and detailed below in order that they may be referred to as needed. We leave it as an exercise for the reader to provide detailed proofs. Proposition 2.9 Let (X, T ) be a flow. Then:
- 43. Flows and their enveloping semigroups 23 (a) The action E(X) × T → E(X) (p, t) → pπt where πt = Rt : E(X) → E(X) q → qt makes E(X, T ) a point transitive flow. (b) The canonical map X : βT → E(X) is both a flow and a semigroup homomorphism. (c) The map E(X) → X p → xp is a flow homomorphism for all x ∈ X. (d) The map βT : βT → E(βT ) is an isomorphism. (e) Let f : (X, T ) → (Y, T ) be a homomorphism of flows. Then f (xp) = f (x)p for all x ∈ X and p ∈ βT . The association of E(X, T ) to a flow (X, T ) is natural in the sense outlined in the following proposition. Proposition 2.10 Let f : (X, T ) → (Y, T ) be a surjective flow homomor- phism. Then there exists a map θ : E(X) → E(Y) such that: (a) the following diagram is commutative βT = βT X ↓ ↓ Y a E(X) θ → E(Y) b ↓ ↓ ↓ ↓ xa X g → Y g(x)b for all homomorphisms g : X → Y and x ∈ X, (b) θ is surjective and continuous, (c) θ(pq) = θ(p)θ(q) for all p, q ∈ E(X) so that θ is both a flow and a semigroup homomorphism, and (d) if ψ : E(X) → E(Y) is a homomorphism with ψ◦ X = Y , then θ = ψ. PROOF: (a) 1. Let a ∈ E(X) and define θ(a) = Y (p) where p ∈ βT with X(p) = a. 2. To see that θ is well-defined, suppose p, q ∈ βT with X(p) = X(q) and let y ∈ Y. 3. Since f is onto, there exists x ∈ X with f (x) = y.
- 44. 24 Flows and their enveloping semigroups 4. y Y (p)= 3 f (x) Y (p) = 2.9(e) f (x X(p))= 2 f (x X(q)) = 2.9(e) f (x) Y (q) = y Y (q). 5. Thus θ is well-defined and θ ◦ X = Y , so the top half of the diagram is commutative. 6. Now let x ∈ X, a ∈ E(X), and p ∈ βT with X(p) = a. 7. Then g(xa) = g(x X(p)) = 2.9(e) g(x) Y (p)= 5 g(x)θ( X(p)) = g(x)θ(a) which shows that the bottom half of the diagram is commutative. (b) 1. θ is surjective because Y is. 2. Let C ⊂ E(Y) be closed. 3. θ−1(C) = X( −1 Y (C)) is closed. ( X, Y are continuous, βT is compact) 4. θ is continuous. (by 2, 3) (c) 1. Let a, b ∈ E(X), and p, q ∈ βT with X(p) = a and X(q) = b. 2. θ(ab) = θ( X(p) X(q)) = 2.9(b) θ( X(pq)) = Y (pq) = 2.9(b) Y (p) Y (q) = θ( X(p))θ( X(q)) = θ(a)θ(b). (d) 1. Assume that ψ : E(X) → E(Y) satisfies ψ ◦ X = Y . 2. For a = X(p), f (xa) = f (x X(p)) = 2.9(e) f (x) Y (p)= 1 f (x)ψ( X(p)) = f (x)ψ(a). 3. f (x)ψ(a) = f (x)θ(a) for all x ∈ X and a ∈ E(X). (by 2 and part (a)) 4. ψ(a) = θ(a) for all a ∈ E(X). (by 2, 3, because f is onto) We end this section with a few examples of how 2.10 is used to identify certain enveloping semigroups. Note that if (X, T ) is a flow, then T acts diagonally on the Cartesian product X × X, so that (x, y)t = (xt, yt), making (X × X, T ) a flow. Corollary 2.11 Let (X, T ) be a flow. Then E(X, T ) ∼ = E(X × X, T ). PROOF: 1. The maps g1 : X × X → X (x, y) → x and g2 : X × X → X are (x, y) → y surjective homomorphisms of flows. 2. There exists a surjective flow and semigroup homomorphism θ : E(X × X) → E(X). (by 1, 2.10) 3. Let p, q ∈ βT with θ( X×X(p)) = θ( X×X(q)). 4. X(p) = X(q). (by 2, 3, 2.10) 5. Let (x, y) ∈ X × X. 6. g1((x, y) X×X(p)) = 2.10 g1(x, y) X(p) = x X(p)= 4 x X(q) = 2.10 g1((x, y) X×X(q)).
- 45. Flows and their enveloping semigroups 25 7. g2((x, y) X×X(p)) = 2.10 g2(x, y) X(p) = y X(p)= 4 y X(q) = 2.10 g2((x, y) X×X(q)). 8. (x, y) X×X(p) = (g1((x, y) X×X(p)), g2((x, y) X×X(p))) = 7 (g1((x, y) X×X(q)), g2((x, y) X×X(q))) = (x, y) X×X(q). 9. X×X(p) = X×X(q). (by 5, 8) 10. θ is an isomorphism. (by 2, 3, 9) Proposition 2.12 Let: (i) (X, T ) be a flow, (ii) x0 ∈ X with x0T = X (so that (X, T ) is point transitive), and (iii) f : E(X) → X be defined by f (p) = x0p for all p ∈ E(X). Then f induces an isomorphism, θ : E(E(X)) ∼ = E(X). PROOF: We leave the proof as an exercise for the reader. Proposition 2.13 Let (X, T ) be a flow. Then E(E(E(X))) ∼ = E(E(X)). PROOF: This follows from 2.12 and the fact that (E(X, T ), T ) is point transitive. EXERCISES FOR CHAPTER 2 Exercise 2.1 (see 2.9) Let: (i) (X, T ) and (Y, T ) be flows, and (ii) f : (X, T ) → (Y, T ) be a homomorphism. Show that (a) The canonical map X : βT → E(X) is both a flow and a semigroup homomorphism. (b) The map E(X) → X p → xp is a flow homomorphism for all x ∈ X. (c) The map βT : βT → E(βT ) is an isomorphism. (d) f (xp) = f (x)p for all x ∈ X and p ∈ βT .
- 46. 26 Flows and their enveloping semigroups Exercise 2.2 (see 2.12) Let: (i) (X, T ) be a flow, (ii) x0 ∈ X with x0T = X (so that (X, T ) is point transitive), and (iii) f : E(X) → X be defined by f (p) = x0p for all p ∈ E(X). Then f induces an isomorphism, θ : E(E(X)) ∼ = E(X). Exercise 2.3 Let: (i) E be a compact Hausdorff space provided with a semigroup structure, (ii) Lp : E → E m → pm for all p ∈ E, (iii) Rp : E → E m → mp for all p ∈ E, and (iv) ϕ : E → EE be defined by ϕ(p) = Rp for all p ∈ E(X). Then (a) ϕ is a semigroup homomorphism, and (b) ϕ is continuous if and only if Lp is continuous for all p ∈ E. In this case ϕ identifies E with a subsemigroup of EE, and E is referred to as an E-semigroup (see also [Akin, E. (1997)]). It is immediate from 2.7, that for any flow (X, T ), its enveloping semigroup E(X, T ) is an E-semigroup.
- 47. 3 Minimal sets and minimal right ideals A subset M ⊂ X of a flow (X, T ) is minimal (see 3.1) if it is a closed non- empty invariant set which is minimal with respect to those properties. One illustration of the interplay between the algebraic and topological properties of the enveloping semigroup is the fact that the minimal subsets of E(X, T ) are exactly the minimal right ideals in E(X, T ) with respect to its semigroup structure. This motivates a study of the algebraic structure of E(X, T ) and its minimal ideals in particular. The key to understanding the structure of a minimal ideal I ⊂ E(X), is an investigation of the idempotents in I (those u ∈ I with u2 = u). In a group, of course, the identity is the only idempotent. We will see that in E(X), any closed sub-semigroup contains an idempotent, so that in general E(X) contains many idempotents in addition to the identity. In particular any right ideal contains an idempotent. In 3.12 we show that if I is a minimal right ideal, then I is a disjoint union {Iv | v ∈ J}, where J = {v ∈ I | v2 = v} is the set of idempotents in I. In fact all of the sets Iv, with v ∈ J are subgroups of I which are isomorphic to one another. We saw in 2.9 that E(βT, T ) ∼ = βT . Thus the preceding discussion applies to any minimal right ideal M in βT . In this case the flow (M, T ) is a universal minimal flow meaning that every minimal flow, is the image of M under some epimorphism. Thus every minimal flow can be identified with the quotient flow (M/R, T ) for some closed, invariant equivalence relation (icer) R on M. This is the basis for our approach to the algebraic theory of minimal flows, in which the structure of M plays a crucial role. This section begins with some background material on minimal sets. We then make explicit the relationship between the minimal sets, and minimal ideals in the enveloping semigroup. We go on to describe the structure of the minimal ideals in E(X, T ), and the section closes with some brief remarks on 27
- 48. 28 Minimal sets and minimal right ideals M which motivate the notation, terminology and approach which will be used in the later sections. Definition 3.1 A subset, M of the flow X is minimal if: (i) ∅ = M, (ii) M is closed, (iii) M is invariant, meaning Mt ⊂ M for all t ∈ T , and (iv) if N ⊂ M satisfies (i), (ii), (iii), then N = M. That is M is minimal with respect to (i),(ii), and (iii). The flow X is minimal (or a minimal set) if X is minimal. Notice that if M is a minimal subset of X, then the flow (M, T ) is a minimal set. We begin with an elementary characterization of minimal sets. This shows that the minimal subsets of a flow reflect its asymptotic properties in the sense that minimality can be characterized in terms of orbit closures. Proposition 3.2 Let: (i) (X, T ) be a flow, and (ii) ∅ = M ⊂ X be closed and invariant. Then the following are equivalent: (a) M is minimal, (b) xT = M for all x ∈ M, and (c) if U ⊂ X is open with M ∩ U = ∅, then M = (U ∩ M)T . PROOF: (a) =⇒ (b) 1. Assume that M is minimal and let x ∈ M. 2. xT ⊂ M satisfies (i), (ii), and (iii) of 3.1. 3. xT = M. (by 1, 2) (b) =⇒ (c) 1. Assume that xT = M for all x ∈ M and let U ⊂ X be open with U ∩ M = ∅. 2. xT ∩ U = ∅ for all x ∈ M. (by 1) 3. x ∈ UT for all x ∈ M. (by 2) (c) =⇒ (a) 1. Assume that (c) holds and N ⊂ M is a nonempty closed, invariant subset of M. 2. Let U = X N. 3. U is open. (by 1, 2) 4. N ∩ (U ∩ M)T ⊂ NT ∩ U = N ∩ U = ∅. (by 1, 2) 5. (U ∩ M)T = M. (by 1, 4)
- 49. Minimal sets and minimal right ideals 29 6. U ∩ M = ∅. (by 1, 5) 7. N = M. (by 1, 2, 6) We use the axiom of choice in the form of Zorn’s lemma both in the following proposition, to show that minimal sets exist, and later to show that certain semigroups contain idempotents. Zorn’s lemma was also used in the appendix to section 1; for the sake of completeness we give a statement here. Zorn’s Lemma 3.3 If (S, ≤) is a partially ordered set such that any increasing chain s1 ≤ · · · ≤ si ≤ · · · has a supremum in S, then S itself has a maximal element. Proposition 3.4 Let (X, T ) be a flow. Then there exists a minimal subset in X. PROOF: 1. Let C = {∅ = C ⊂ X | C is closed and invariant}. 2. C is partially ordered by the relation C1 ⊃ C2. 3. Let = {C1 ⊃ C2 ⊃ · · · Ci ⊃ · · · } be an increasing chain of elements of C. 4. C = Ci is nonempty, closed and invariant, hence C ∈ C is a supremum for . (X is compact) 5. C has a maximal element M ⊂ X. (by Zorn’s lemma) 6. M is a minimal subset of X. (by 1, 2, 5) It is an elementary but important fact, detailed in the next proposition, that minimality is preserved by homomorphisms. Proposition 3.5 Let ϕ : X → Y be a flow homomorphism. (a) If M is a minimal subset of X, then ϕ(M) is a minimal subset of Y. (b) If N is a minimal subset of ϕ(X), then there exists a minimal subset M of X with ϕ(M) = N. PROOF: (a) 1. Let K be a non-empty closed invariant subset of ϕ(M). 2. ϕ−1(K) ∩ M is a closed non-empty invariant subset of the minimal set M. 3. ϕ−1(K) ∩ M = M. 4. K = ϕ(M). (by 3) (b) 1. ϕ−1(N) is a non-empty closed invariant subset of X. 2. There exists a minimal subset M of ϕ−1(N). (by 3.4) 3. ϕ(M) is a minimal subset of the minimal set N, so ϕ(M) = N. (by (a)) Now we turn to a discussion of the minimal ideals in the enveloping semigroup E(X, T ) of a flow (X, T ). We first show that they coincide with the minimal sets of the flow (E(X, T ), T ); then we will describe the structure of these minimal ideals.
- 50. 30 Minimal sets and minimal right ideals Definition 3.6 Let X be a flow, and E its enveloping semigroup. Then a nonempty subset I of E is a (right) ideal if IE ⊂ I. The ideal is minimal if it contains no ideals as proper subsets. Note that if I ⊂ E(X, T ) is an ideal, then IT ⊂ I, whence if I is closed, (I, T ) is a flow. In fact as the following proposition shows, (I, T ) is a minimal flow. Proposition 3.7 Let: (i) (X, T ) be a flow, (ii) E = E(X, T ), and (iii) I an ideal in E. Then I is a minimal ideal if and only if I is closed and the flow (I, T ) is minimal. PROOF: =⇒ 1. Assume that I is minimal and let p ∈ I. 2. pT = pE ⊂ I. (I is an ideal) 3. Lp(E) = pE = I. (pE is an ideal and I is minimal) 4. I is closed. (by 3, E is compact and Lp is continuous by 2.7) 5. (I, T ) is minimal. (by 2, 3, 3.2) ⇐= 1. Assume that (I, T ) is a minimal flow, let J ⊂ I be an ideal, and p ∈ J. 2. I = pT = pE ⊂ J. 3. I = J. (by 1, 2) Corollary 3.8 Let: (i) (X, T ) be a flow, (ii) E = E(X, T ), and (iii) I an ideal in E. Then I contains a minimal ideal. PROOF: This follows from 3.7 and 3.4; we leave the details to the reader. The description of the structure of a minimal ideal I ⊂ E(X, T ) relies on the existence of idempotents u2 = u ∈ I. Thus the importance of the following theorem, which is an interesting example of the interplay between the topolog- ical and algebraic structure in a topological space which is also a semigroup. Theorem 3.9 Let X be a compact T1 (single points are closed) semigroup such that the left-multiplication maps
- 51. Minimal sets and minimal right ideals 31 Lx : X → X y → xy are continuous and closed for all x ∈ X. Then there exists an idempotent u in X. (u is an idempotent if u2 = u.) PROOF: 1. Let = {S ⊂ X | ∅ = S = S and S2 ⊂ S}. 2. Then = ∅ and by Zorn’s lemma there exists a minimal element, S of when the latter is ordered by inclusion. 3. Let s ∈ S. 4. sS = Ls(S) is a closed subset of S. (by 1, 2, 3, Ls is closed) 5. sSsS ⊂ sSS ⊂ sS ⊂ S, whence sS = S by the minimality of S. 6. Let R = {t ∈ S | st = s}. 7. ∅ = R. (by 3, 5) 8. R2 ⊂ R = L−1 s {s} = R. (Ls is continuous, X is T1) 9. R = S. (by 2, 6, 7, 8) 10. s2 = s (by 3, 6, 9) Since every continuous map from a compact space to a Hausdorff space is a closed map, the following is an immediate consequence of 3.9. Corollary 3.10 Let X be a compact Hausdorff semigroup such that the maps Lx : X → X are continuous for all x ∈ X. (We often refer to such a semigroup as an E-semigroup, see 2.E.3.) Then there exists an idempotent u in X. Corollary 3.11 Let X be a compact T1 group such that left multiplication is continuous, and let S be a closed sub-semigroup of X. Then S is a subgroup of X. PROOF: 1. Let x ∈ S. 2. xS = L−1 x−1 (S) is a closed subsemigroup of S. (Lx−1 is continuous) 3. There exists an idempotent u ∈ xS. (by 1, 2, 3.9) 4. u = id. (X is a group) 5. x−1 ∈ S. (by 3, 4) For any flow (X, T ), the enveloping semigroup E(X, T ) is an E-semigroup in the sense of 3.10, and hence any minimal ideal I ⊂ E(X) is also an E-semigroup and therefore contains idempotents. It turns out that each of the idempotents in I acts as a left-identity on I. In fact the ideal I can be
- 52. 32 Minimal sets and minimal right ideals partitioned into a disjoint union of groups, each one containing exactly one idempotent (which serves as the identity for that group). We give the details in the following theorem. Theorem 3.12 Let: (i) (X, T ) be a flow, (ii) E = E(X, T ), and (iii) I ⊂ E be a minimal ideal in E. Then: (a) The set J of idempotents of I is non-empty, (b) vp = p for all v ∈ J and p ∈ I, (c) Iv is a group with identity v, for all v ∈ J, (d) {Iv | v ∈ J} is a partition of I, and (e) if we set G = Iu for some u ∈ J, then I = {Gv | v ∈ J} (disjoint union). PROOF: (a) 1. This follows immediately from 3.10. (b) 1. Let v ∈ J, and p ∈ I. 2. vI ⊂ I. (I is an ideal) 3. vI = I. (by 2, I is minimal) 4. There exists q ∈ I with vq = p. (by 3) 5. vp = 4 vvq = vq = 4 p. (c) 1. Let q ∈ Iv. 2. There exists p ∈ I with q = pv. 3. qv = pvv = pv = q. 4. v is both a left and right identity for Iv. (by (b), and 3) 5. qI is an ideal. (I is an ideal) 6. qI = I. (I is minimal) 7. There exists q ∈ I with qq = v. (by 6) 8. q(q v) = (qq )v = vv = v. (by 7) 9. (q q)(q q) = q (qq )q = 7 q (vq) = (b) q q. 10. (q v)q = (b) q q = 3 (q q)v = 9,(b) v. 11. q v is a left and right inverse of q in Iv. (by 8, 10) (d) 1. Let p ∈ I. 2. As before pI = I. (I is a minimal ideal) 3. K = {q ∈ I | pq = p} = L−1 p (p) is a nonempty closed subsemigroup of I. (by 2, 2.9) 4. There exists an idempotent u ∈ J with pu = p. (by 3.9) 5. p ∈ Iu. (by 4)
- 53. Minimal sets and minimal right ideals 33 6. I = {Iv | v ∈ J}. (by 1, 5) 7. Finally let u, v ∈ J and p ∈ Iv ∩ Iu. 8. p = pu = pv and there exists q ∈ Iv with qp = v. (by 8, (c)) 9. u = (b) vu = (qp)u = q(pu) = 8 qp = 8 v. (e) This is just a restatement of (d). Note that each of the groups Iv in proposition 3.12 is isomorphic to the group G = Iu; indeed the map pu → pv is an isomorphism since (pv)(qv) = p(vq)v = (pq)v. It is interesting to note that given an abstract group G and an index set J, one can define a semigroup structure on the disjoint union I = v∈J Gv of copies of G, in which J can be identified with the set of idempotents. We simply define pvqw = (pq)w ∈ Gw for p, q ∈ G and v, w ∈ J. Then {ev | v ∈ J} is the set of idempotents in I. Proposition 3.12 shows that every minimal ideal in E(X, T ) has this structure. For any semigroup E one can define an equivalence relation on the set J of idempotents in E as follows: u ∼ v ⇐⇒ uv = u and vu = v. This relation is clearly reflexive and symmetric; to check transitivity we observe that: u ∼ v ∼ w ⇒ uw = (uv)w = u(vw) = uv = u and wu = (wv)u = w(vu) = wv = w. This motivates the following definition. Definition 3.13 Let (X, T ) be a flow. An idempotent u2 = u ∈ E(X, T ) is said to be a minimal idempotent if u is contained in some minimal ideal I ⊂ E(X, T ). If u, v ∈ E(X, T ) are idempotents with uv = u and vu = v, we write u ∼ v and refer to u and v as equivalent idempotents. When u and v are in the same minimal ideal I, then u ∼ v ⇒ u = v since by 3.12 both u and v act as left-identities on I. Thus for any minimal idempotent, the equivalence class [u] intersects each minimal ideal at most once; we leave it as an exercise for the reader to check that [u] contains only minimal idempotents. The following proposition says that [u] intersects every minimal ideal exactly once, so that [u] consists of one idempotent from every minimal ideal in E(X, T ). Proposition 3.14 Let: (i) (X, T ) be a flow, (ii) E = E(X, T ),
- 54. 34 Minimal sets and minimal right ideals (iii) I, K ⊂ E be minimal ideals in E, and (iv) u2 = u ∈ I be an idempotent. Then there exists a unique idempotent v ∈ K with uv = u and vu = v. PROOF: 1. Let u2 = u ∈ I. 2. uK is a closed ideal in I, whence uK = I. 3. N ≡ {k ∈ K | uk = u} = ∅. (by 2) 4. N = L−1 u (u) ∩ K is closed, and N2 ⊂ N. 5. There exists v2 = v ∈ N. (by 4, 3.10) 6. uv = u. (by 3, 5) 7. Similarly there exists w2 = w ∈ I with vw = v. (applying 1-6 to v ∈ K) 8. w = 1,3.12 uw = 6 uvw = 7 uv = 6 u. 9. vu = v. (by 7, 8) 10. Now suppose η2 = η ∈ K with uη = u and ηu = η. 11. η = 10 ηu = 6 ηuv = 10 ηv. 12. η ∈ Kη ∩ Kv. (by 10, 11) 13. η = v. (by 3.12) As an immediate consequence of 3.12 we see that any two minimal ideals in E(X, T ) are isomorphic as minimal flows in a natural way. Proposition 3.15 Let: (i) (X, T ) be a flow, (ii) E = E(X, T ), (iii) I, K ⊂ E be minimal ideals in E, (iv) u2 = u ∈ I be an idempotent, and (v) v2 = v ∈ K with u ∼ v. Then the map Lv : (I, T ) → (K, T ) p → vp is an isomorphism, its inverse being the map Lu. PROOF: We leave the proof as an exercise for the reader. The structure of the minimal ideals in the enveloping semigroup E(X, T ) of any flow (X, T ) described above, and the minimal idempotents themselves play an important role (see in particular section 4) in the study of the dynamics of (X, T ). The minimal ideals in the semigroup βT have exactly the same structure. This can be seen by noting that the proofs of 3.12, 3.14, and 3.15 rely only on properties of E(X, T ) which are shared by βT . On the other hand we saw in 2.9, that E(βT ) ∼ = βT as semigroups. Thus 3.12, 3.14, and 3.15 can be applied directly to βT . In particular, we can speak of minimal idempotents
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. in bills, and quite as sadly deposited, in addition, his favorite chronometer. The highwayman picked up the watch, looked it over, shook his head and, thanking Ben, returned it, expressing the hope that, whatever adversity might overwhelm him, he should never be discovered with such a timepiece! All in all, the robbers secured nearly two thousand dollars; but, strange to relate, they overlooked the treasure in the Wells Fargo chest, as well as several hundred dollars in greenbacks belonging to the Government. Sheriff J. F. Burns and Deputy H. C. Wiley pursued and captured the robbers; and within about a week they were sent to the Penitentiary. On the same evening, at high tide, the little steamer christened Los Angeles and constructed by P. Banning Company to run from the wharf to the outside anchorage, was committed to the waters, bon- fires illuminating quite distinctly both guests and the neighboring landscape, and lending to the scene a weird and charming effect. In a previous chapter I have given an account of Lady Franklin's visit to San Pedro and Los Angeles, and of the attention shown her. Her presence awakened new interest in the search for her lamented husband, and paved the way for the sympathetic reception of any intelligence likely to clear up the mystery. No little excitement, therefore, was occasioned eight years later by the finding of a document at San Buenaventura that seemed like a voice from the dead. According to the story told, as James Daly (of the lumber firm of Daly Rodgers) was walking on the beach on August 30th, he found a sheet of paper a foot square, much mutilated but bearing, in five or six different languages, a still legible request to forward the memoranda to the nearest British Consul or the Admiralty at London. Every square inch of the paper was covered with data relating to Sir John Franklin and his party, concluding with the definite statement that Franklin had died on June 11th, 1847. Having been found within a week of the time that the remnant of Dr. Hall's party, which went in search of the explorer, had arrived home in Connecticut with the announcement that they had discovered seven skeletons of Franklin's men, this document, washed up on the Pacific Coast,
- 57. excited much comment; but I am unable to say whether it was ever accepted by competent judges as having been written by Franklin's associates. In 1869, the long-familiar adobe of José António Carrillo was razed to make way for what, for many years, was the leading hotel of Los Angeles. This was the Pico House, in its decline known as the National Hotel, which, when erected on Main Street opposite the Plaza at a cost of nearly fifty thousand dollars, but emphasized in its contrasting showiness the ugliness of the neglected square. Some thirty-five thousand dollars were spent in furnishing the eighty-odd rooms, and no little splurge was made that guests could there enjoy the luxuries of both gas and baths! In its palmy days, the Pico House welcomed from time to time travelers of wide distinction; while many a pioneer, among them not a few newly-wedded couples now permanently identified with Los Angeles or the Southland, look back to the hostelry as the one surviving building fondly associated with the olden days. Charles Knowlton was an early manager; and he was succeeded by Dunham Schieffelin. Competition in the blacking of boots enlivened the fall, the Hotel Lafayette putting boldly in printer's ink the question, Do You Want to Have Your Boots Blacked in a Cool, Private Place? This challenge was answered with the following proclamation: Champion Boot-Black! Boots Blacked Neater and Cheaper than Anywhere Else in the City, at the Blue Wing Shaving Saloon by D. Jefferson. Brickmaking had become, by September, quite an important industry. Joe Mullally, whose brickyard was near the Jewish Cemetery, then had two kilns with a capacity of two hundred and twenty-five thousand; and in the following month he made over five hundred thousand brick. In course of time, the Los Angeles San Pedro Railroad was completed to the Madigan lot, which remained for several years the Los Angeles terminus; and justly confident that the difficulty with the
- 58. authorities would be removed, the Company pushed work on their depot and put in a turn-table at the foot of New Commercial Street. There was but one diminutive locomotive, though a larger one was on its way around the Horn from the East and still another was coming by the Continental Railway; and every few days the little engine would go out of commission, so that traffic was constantly interrupted. At such times, confidence in the enterprise was somewhat shaken; but new rolling stock served to reassure the public. A brightly-painted smoking-car, with seats mounted on springs, was soon the talk of the town. I have spoken of J. J. Reynolds's early enterprise and the competition that he evoked. Toward the end of July, he went up to San Francisco and outdid Hewitt by purchasing a handsome omnibus, suitable for hotel service and also adapted to the needs of families or individuals clubbing together for picnics and excursions. This gave the first impetus to the use of hotel 'buses, and by the first Sunday in September, when the cars from Wilmington rolled in bringing passengers from the steamer Orizaba, the travelers were met by omnibuses and coaches from all three hotels, the Bella Union, the United States and the Lafayette; the number of vehicles, public and private, giving the streets around the railroad depot a very lively appearance. Judge W. G. Dryden, so long a unique figure here, died on September 10th and A. J. King succeeded him as County Judge. A notable visit to Los Angeles was that of Secretary William H. Seward who, in 1869, made a trip across the Continent, going as far north as Alaska and as far south as Mexico, and being everywhere enthusiastically received. When Seward left San Francisco for San Diego, about the middle of September, he was accompanied by Frederick Seward and wife (his son and daughter-in-law), General W. S. Rosecrans, General Morton C. Hunter, Colonel Thomas Sedgwick and Senator S. B. Axtell; and the news of their departure having been telegraphed ahead, many people went down to greet them on the arrival of the steamer Orizaba. After the little steamer Los
- 59. Angeles had been made fast to the wharf, it was announced, to everyone's disappointment, that the Secretary was not coming ashore, as he wished to continue on his way to San Diego. Meanwhile, the Common Council had resolved to extend the hospitality of the City to the distinguished party; and by September 19th, posters proclaimed that Seward and his party were coming and that citizens generally would be afforded an opportunity to participate in a public reception at the Bella Union on September 21st. A day in advance, therefore, the Mayor and a Committee from the Council set out for Anaheim, where they met the distinguished statesman on his way, whence the party jogged along leisurely in a carriage and four until they arrived at the bank of the Los Angeles River; and there Seward and his friends were met by other officials and a cavalcade of eighty citizens led by the military band of Drum Barracks. The guests alighted at the Bella Union and in a few minutes a rapidly-increasing crowd was calling loudly for Mr. Seward. The Secretary, being welcomed on the balcony by Mayor Joel H. Turner, said that he had been laboring under mistakes all his life: he had visited Rome to witness celebrated ruins, but he found more interesting ruins in the Spanish Missions (great cheers); he had journeyed to Switzerland to view its glaciers, but upon the Pacific Coast he had seen rivers of ice two hundred and fifty feet in breadth, five miles long and God knows how high (more cheers); he had explored Labrador to examine the fisheries, but in Alaska he found that the fisheries came to him (Hear! hear! and renewed applause); he had gone to Burgundy to view the most celebrated vineyards of the world, but the vineyards of California far surpassed them all! (Vociferous and deafening hurrahs, and tossing of bouquets.) The next day the Washington guests and their friends were shown about the neighborhood, and that evening Mr. Seward made another and equally happy speech to the audience drawn to the Bella Union by the playing of the band. There were also addresses by the Mayor, Senator Axtell, ex-Governor Downey and others, after which, in good
- 60. old American fashion, citizens generally were introduced to the associate of the martyred Lincoln. At nine o'clock, a number of invited guests were ushered into the Bella Union's dining-room where, at a bounteous repast, the company drank to the health of the Secretary. This brought from the visitor an eloquent response with interesting local allusions. Secretary Seward remarked that he found people here agitated upon the question of internal improvements—for everywhere people wanted railroads. Californians, if they were patient, would yet witness a railroad through the North, another by the Southern route, still another by the Thirty-fifth parallel, a fourth by the central route, and lastly, as the old plantation song goes, one down the middle! California needed more population, and railroads were the means by which to get people. Finally, Mr. Seward spoke of the future prospects of the United States, saying much of peculiar interest in the light of later developments. We were already great, he affirmed; but a nation satisfied with its greatness is a nation without a future. We should expand, and as mightily as we could; until at length we had both the right and the power to move our armies anywhere in North America. As to the island lying almost within a stone's throw of our mainland, ought we not to possess Cuba, too? Other toasts, such as The Mayor and Common Council, The Pioneers, The Ancient Hospitality of California, The Press, The Wine Press and Our Wives and Sweethearts, were proposed and responded to, much good feeling prevailing notwithstanding the variance in political sentiments represented by guests and hosts; and everyone went home, in the small hours of the morning, pleased with the manner in which Los Angeles had received her illustrious visitors. The next day, Secretary Seward and party left for the North by carriages, rolling away toward Santa Barbara and the mountains so soon to be invaded by the puffing, screeching iron horse.
- 61. Recollecting this banquet to Secretary Seward, I may add an amusing fact of a personal nature. Eugene Meyer and I arranged to go to the dinner together, agreeing that we were to meet at the store of S. Lazard Company, almost directly opposite the Bella Union. When I left Los Angeles in 1867, evening dress was uncommon; but in New York I had become accustomed to its more frequent use. Rather naturally, therefore, I donned my swallowtail; Meyer, however, I found in a business suit and surprised at my query as to whether he intended going home to dress? Just as we were, we walked across the street and, entering the hotel, whom should we meet but ex-Mayor John G. Nichols, wearing a grayish linen duster, popular in those days, that extended to his very ankles; while Pio and Andrés Pico came attired in blue coats with big brass buttons. Meyer, observing the Mayor's outfit, facetiously asked me if I still wished him to go home and dress according to Los Angeles fashion; whereupon I drew off my gloves, buttoned up my overcoat and determined to sit out the banquet with my claw-hammer thus concealed. Mr. Seward, it is needless to say, was faultlessly attired. The Spanish archives were long neglected, until M. Kremer was authorized to overhaul and arrange the documents; and even then it was not until September 16th that the Council built a vault for the preservation of the official papers. Two years later, Kremer discovered an original proclamation of peace between the United States and Mexico. Elsewhere I allude to the slow development of Fort Street. For the first time, on the twenty-fourth of September street lamps burned there, and that was from six to nine months after darkness had been partially banished from Nigger Alley, Los Angeles, Aliso and Alameda streets.
- 62. Phineas Banning, about 1869
- 63. Henri Penelon, in his Studio
- 64. Carreta, Earliest Mode of Transportation Alameda Street Depot and Train, Los Angeles San Pedro Railroad Supplementing what I have said of the Los Angeles San Pedro Railroad depot: it was built on a lot fronting three hundred feet on Alameda Street and having a depth of one hundred and twenty feet, its situation being such that, after the extension of Commercial Street, the structure occupied the southwest corner of the two highways. Really, it was more of a freight-shed than anything else, without adequate passenger facilities; a small space at the North end contained a second story in which some of the clerks slept; and in a cramped little cage beneath, tickets were sold. By the way, the engineer of the first train to run through to this depot was James Holmes, although B. W. Colling ran the first train stopping inside the city limits. About this time the real estate excitement had become still more intense. In anticipation of the erection of this depot, Commercial Street property boomed and the first realty agents of whom I have any recollection appeared on the scene, Judge R. M. Widney being among them. I remember that two lots—one eighty by one hundred
- 65. and twenty feet in size at the northwest corner of First and Spring streets, and the other having a frontage of only twenty feet on New Commercial Street, adjacent to the station—were offered simultaneously at twelve hundred dollars each. Contrary, no doubt, to what he would do to-day, the purchaser chose the Commercial Street lot, believing that location to have the better future. Telegraph rates were not very favorable, in 1869, to frequent or verbose communication. Ten words sent from Los Angeles to San Francisco cost one dollar and a half; and fifty cents additional was asked for the next five words. After a while, there was a reduction of twenty-five per cent, in the cost of the first ten words, and fifty per cent, on the second five. Twenty-four hundred voters registered in Los Angeles this year. In the fall, William H. Spurgeon founded Santa Ana some five miles beyond Anaheim on a tract of about fifty acres, where a number of the first settlers experimented in growing flax. It is not clear to me just when the rocky Arroyo Seco began to be popular as a resort, but I remember going there on picnics as early as 1857. By the late sixties, when Santa Monica Cañon also appealed to the lovers of sylvan life, the Arroyo had become known as Sycamore Grove—a name doubtless suggested by the numerous sycamores there—and Clois F. Henrickson had opened an establishment including a little hotel, a dancing-pavilion, a saloon and a shooting-alley. Free lunch and free beer were provided for the first day, and each Sunday thereafter in the summer season an omnibus ran every two hours from Los Angeles to the Sycamores. After some years, John Rumph and wife succeeded to the management, Frau Rumph being a popular Wirtin; and then the Los Angeles Turnverein used the grove for its public performances, including gymnastics, singing and the old-time sack-racing and target-shooting.
- 66. James Miller Guinn, who had come to California in November, 1863 and had spent several years in various counties of the State digging for gold and teaching school, drifted down to Los Angeles in October and was soon engaged as Principal of the public school at the new town of Anaheim, remaining there in that capacity for twelve years, during part of which time he also did good work on the County School Board. Under the auspices of the French Benevolent Society and toward the end of October, the corner-stone of the French Hospital built on City donation lots, and for many years and even now one of the most efficient institutions of our city, was laid with the usual ceremonies. On October 9th, the first of the new locomotives arrived at Wilmington and a week later made the first trial trip, with a baggage and passenger car. Just before departure a painter was employed to label the engine and decorate it with a few scrolls; when it was discovered, too late, that the artist had spelled the name: LOS ANGELOS. On October 23d, two lodges of Odd Fellows used the railway to visit Bohen Lodge at Wilmington, returning on the first train, up to that time, run into Los Angeles at midnight. October 26th was a memorable day, for on that date the Los Angeles San Pedro Railroad Company opened the line to the public and invited everybody to enjoy a free excursion to the harbor. Two trains were dispatched each way, the second consisting of ten cars; and not less than fifteen hundred persons made the round trip. Unfortunately, it was very warm and dusty, but such discomforts were soon forgotten in the novelty of the experience. On the last trip back came the musicians; and the new Los Angeles depot having been cleared, cleaned up and decorated for a dedicatory ball, there was a stampede to the little structure, filling it in a jiffy. Judge H. K. S. O'Melveny, who first crossed the Plains from Illinois on horseback in 1849, came to Los Angeles with his family in November, having already served four years as a Circuit Judge, following his practice of law in Sacramento. He was a brother-in-law
- 67. of L. J. Rose, having married, in 1850, Miss Annie Wilhelmina Rose. Upon his arrival, he purchased the southwest corner of Second and Fort streets, a lot one hundred and twenty by one hundred and sixty-five feet in size, and there he subsequently constructed one of the fine houses of the period; which was bought, some years later, by Jotham Bixby for about forty-five hundred dollars, after it had passed through various hands. Bixby lived in it for a number of years and then resold it. In 1872, O'Melveny was elected Judge of Los Angeles County; and in 1887, he was appointed Superior Judge. H. W. O'Melveny, his second son, came from the East with his parents, graduating in time from the Los Angeles High School and the State University. Now he is a distinguished attorney and occupies a leading position as a public-spirited citizen, and a patron of the arts and sciences. In his very readable work, From East Prussia to the Golden Gate, Frank Lecouvreur credits me with having served the commonwealth as Supervisor. This is a slight mistake: I was an unwilling candidate, but never assumed the responsibilities of office. In 1869, various friends waited upon me and requested me to stand as their candidate for the supervisorship; to which I answered that I would be glad to serve my district, but that I would not lift a finger toward securing my election. H. Ábila was chosen with six hundred and thirty-one votes, E. M. Sanford being a close second with six hundred and sixteen; while five hundred and thirty-seven votes were cast in my favor. Trains on the new railway began to run regularly on November 1st; and there still exists one of the first time-tables, bearing at the head, Los Angeles San Pedro Railroad and a little picture of a locomotive and train. At first, the train scheduled for two stated round trips a day (except on steamer days, when the time was conditioned by the arrival and departure of vessels) left Wilmington at eight o'clock in the morning and at one o'clock in the afternoon, returning at ten in the morning and four in the afternoon. The fare between Los Angeles and Wilmington was one dollar and fifty cents,
- 68. with an additional charge of one dollar to the Anchorage; while on freight from the Anchorage to Los Angeles, the tariff was: dry goods, sixteen dollars per ton; groceries and other merchandise, five dollars; and lumber, seven dollars per thousand feet. After the formal opening of the railroad, a permanent staff of officers, crew and mechanicians was organized. The first Superintendent was H. W. Hawthorne, who was succeeded by E. E. Hewitt, editor of the Wilmington Journal. N. A. McDonald, was the first conductor; Sam Butler was the first and, for a while, the only brakeman, and the engineers were James McBride and Bill Thomas. The first local agent was John Milner; the first agent at Wilmington, John McCrea. The former was succeeded by John E. Jackson, who from 1880 to 1882 served the community as City Surveyor. Worthy of remark, perhaps, as a coincidence, is the fact that both Milner and McCrea ultimately became connected in important capacities with the Farmers Merchants Bank. The first advertised public excursion on the Los Angeles San Pedro Railroad after its opening was a trip to Wilmington and around San Pedro Harbor, arranged for November 5th, 1869. The cars, drawn by the locomotive Los Angeles and connecting with the little steamer of the same name, left at ten and returned at three o'clock in the afternoon. Two dollars was the round-trip fare, while another dollar was exacted from those who went out upon the harbor. In the late seventies, a Portuguese named Fayal settled near what is now the corner of Sixth and Front streets, San Pedro; and one Lindskow took up his abode in another shack a block away. Around these rude huts sprang up the neighborhoods of Fayal and Lindville, since absorbed by San Pedro. Probably the first attempt to organize a fire company for Los Angeles was made in 1869, when a meeting was called on Saturday evening, November 6th, at Buffum's Saloon, to consider the matter. A temporary organization was formed, with Henry Wartenberg as President; W. A. Mix, Vice-President; George M. Fall, Secretary; and
- 69. John H. Gregory, Treasurer. An initiation fee of two dollars and a half, and monthly dues of twenty-five cents, were decided upon; and J. F. Burns, B. Katz, Emil Harris, George Pridham, E. B. Frink, C. D. Hathaway, P. Thompson, O. W. Potter, C. M. Small and E. C. Phelps were charter members. A committee appointed to canvass for subscriptions made little progress, and the partial destruction of Rowan's American Bakery, in December, demonstrating the need of an engine and hose cart, brought out sharp criticism of Los Angeles's penuriousness. About the middle of November, Daniel Desmond, who had come on October 14th of the preceding year, opened a hat store on Los Angeles Street near New Commercial, widely advertising the enterprise as a pioneer one and declaring, perhaps unconscious of any pun, that he proposed to fill a want that had long been felt. The steamer Orizaba, which was to bring down Desmond's goods, as ill luck would have it left half of his stock lying on the San Francisco pier; and the opening, so much heralded, had to be deferred several weeks. As late as 1876, he was still the only exclusive hatter here. Desmond died on January 23d, 1903, aged seventy years, and was succeeded by his son, C. C. Desmond. Another son, D. J. Desmond, is the well-known contractor. Toward the close of November, Joseph Joly, a Frenchman, opened the Chartres Coffee Factory on Main Street opposite the Plaza, and was the pioneer in that line. He delivered to both stores and families, and for a while seemed phenomenally successful; but one fine morning in December it was discovered that the Jolly Joseph had absconded, leaving behind numerous unpaid bills. The first marble-cutter to open a workshop in Los Angeles was named Miller. He came toward the end of 1869 and established himself in the Downey Block. Prior to Miller's coming, all marble work was brought from San Francisco or some source still farther away, and the delay and expense debarred many from using that stone even for the pious purpose of identifying graves.
- 70. With the growth of Anaheim as the business center of the country between the new San Gabriel and the Santa Ana rivers, sentiment had been spreading in favor of the division of Los Angeles County; and at the opening of the Legislature of 1869-70, Anaheim had its official representative in Sacramento, ready to present the claims of the little German settlement and its thriving neighbors. The person selected for this important embassy was Major Max von Stroble; and he inaugurated his campaign with such sagacity and energy that the bill passed the Assembly and everything pointed to an early realization of the scheme. It was not, however, until Los Angeles awoke to the fact that the proposed segregation meant a decided loss, that opposition developed in the Senate and the whole matter was held up. Stroble thereupon sent posthaste to his supporters for more cash, and efforts were made to get the stubborn Senate to reconsider. Doubtless somebody else had a longer purse than Stroble; for in the end he was defeated, and the German's dream did not come true until long after he had migrated to the realms that know no subdivisions. One of the arguments used in favor of the separation was that it took two days's time, and cost six dollars, for the round trip to the Los Angeles Courthouse; while another contention then regarded as of great importance was that the one coil of hose pipe owned by the County was kept at Los Angeles! Stroble, by-the-way, desired to call the new county Anaheim. Major von Stroble was a very interesting character. He was a German who had stood shoulder to shoulder with Carl Schurz and Franz Sigel in the German Revolution of 1848, and who, after having taken part in the adventures of Walker's filibustering expedition to Nicaragua, finally landed in Anaheim, where he turned his attention to the making of wine. He soon tired of that, and in 1867 was found boring for oil on the Brea Ranch, again meeting with reverses where others later were so successful. He then started the movement to divide Los Angeles County and once more failed in what was afterward accomplished. Journalism in Anaheim next absorbed him and, having
- 71. had the best of educational advantages, Stroble brought to his newspaper both culture and the experience of travel. The last grand effort of this adventurous spirit was the attempt to sell Santa Catalina Island. Backed by the owners, Stroble sailed for Europe and opened headquarters near Threadneedle Street in London. In a few weeks he had almost effected the sale, the contract having been drawn and the time actually set for the following day when the money—a cool two hundred thousand pounds—was to be paid; but no Stroble kept tryst to carry out his part of the transaction. Only the evening before, alone and unattended, the old man had died in his room at the very moment when Fortune, for the first time, was to smile upon him! Eighteen or twenty years later, Catalina was sold for much less than the price once agreed upon.
- 72. CHAPTER XXVIII THE LAST OF THE VIGILANTES 1870 As I have somewhere related, I began buying hides as far back as 1855, but it was not until 1870 that this branch of our business assumed such importance as to require more convenient quarters. Then we bought a place on the southeast corner of Alameda and Commercial streets, facing sixty feet on Alameda and having a depth of one hundred and sixty-five feet, where we constructed a hide- house and erected a press for baling. We paid P. Beaudry eleven hundred dollars for the lot. The relatively high price shows what the Los Angeles San Pedro Railroad depot had done for that section. In the days when hides were sent by sailing-vessels to the East, a different method of preparing them for shipment was in vogue. The wet hides having been stretched, small stakes were driven into the ground along the edge of, and through the skins, thus holding them in place until they had dried and expanding them by about one- third; in this condition they were forwarded loose. Now that transportation is more rapid and there are tanneries in California, all hides are handled wet. In 1870, business life was centered on Los Angeles Street between Commercial and Arcadia; and all the hotels were north of First Street. Fort Street ended in a little bluff at a spot now between
- 73. Franklin and First streets. Spring Street was beginning to take on new life, and yet there was but one gas lamp along the entire roadway, though many were the appeals to add another lamp, say, as far as First Street! Sometime in January, a number of ladies of this city met and, through the exertions of Mrs. Rosa Newmark, wife of Joseph Newmark, formed the Ladies' Hebrew Benevolent Society. Mrs. Newmark, as was once pointed out in a notable open-air meeting of women's clubs (to which I elsewhere refer), never accepted any office in the Society; but for years she was untiring in her efforts in the cause of charity. The first officers were: President, Mrs. W. Kalisher; Vice-President, Mrs. Harris Newmark; Treasurer, Mrs. John Jones; Secretary, Mrs. B. Katz; and Collector, Mrs. A. Baer. Three Counselors—Henry Wartenberg, I. M. Hellman and myself— occasionally met with the ladies to advise them. Aside from the fact of its importance as the pioneer ladies' benevolent organization instituted in Los Angeles, the Society found a much-needed work to do. It was then almost impossible to obtain nurses, and the duty devolved on members to act in that capacity, where such assistance was required, whether the afflicted were rich or poor. It was also their function to prepare the dead for interment, and to keep proper vigil over the remains until the time of burial. During the year 1869 or 1870, as the result of occasional gatherings in the office of Dr. Joseph Kurtz, the Los Angeles Turnverein was organized with eleven members—Emil Harris leading in the movement, assisted by Dr. Kurtz, Ed. Preuss, Lorenzo Leck, Philip and Henry Stoll, Jake Kuhrts, Fred Morsch, C. C. Lips and Isaac Cohn. Dr. Kurtz was elected President. They fraternized for a while at Frau Wiebecke's Garden, on the west side of Alameda near First Street, about where the Union Hardware and Metal Company now stands; and there, while beer and wine were served in the open air, the Teutons gratified their love of music and song. Needing for their gymnastics more enclosed quarters, the Turnverein rented of Kalisher Wartenberg the barn on Alameda Street between
- 74. Ducommon and First, used as a hide-house; and in that rough- boarded shack, whose none too aromatic odors are still a souvenir to many a pioneer resident, the Turners swung and vaulted to their heart's content. Classes were soon arranged for boys; and the envy of all was the lad who, after numerous risks to limb and neck, proudly topped the human pyramid. Another garden of this period often patronized by the Turnverein was Kiln Messer's, on First Street between Alameda and the river. The Post Office was moved this year from the corner of North Main and Market streets to the middle of Temple Block, but even there the facilities were so inadequate that Wells Fargo Company, in June, put up a letter-box at the corner of Main and Commercial streets which was emptied but once a day, at four o'clock in the afternoon, save on steamer days when letters were taken out at half-past nine. One other box was at the sole railroad depot, then at the corner of Alameda and Commercial streets. The Post Office at that time was also so miserably illuminated that citizens fumbled about to find their letter-boxes, and ladies were timid about entering the building at night. Postmasters were allowed small reserves; and for some time in 1870 the Los Angeles Post Office was entirely out of one- and two-cent stamps. In February, the way was prepared for the first city directory when the houses of Los Angeles were ordered to be numbered, a public discussion of the need for a directory having taken place the previous December. When the collaborators began to collect names and other data, there were many refusals to answer questions; but the little volume of seventy pages was finally published in 1871. Until 1870 Los Angeles had no bookbinder, all binding having had to be sent to San Francisco; and a call was then sent out to induce a journeyman to settle here. On the fourteenth of February, Phineas Banning was married to Miss Mary, daughter of Colonel J. H. Hollister—the affair being the consummation of a series of courtly addresses in which, as I have
- 75. related, it was my pleasurable privilege to play an intermediary part. As might be expected of one who was himself an experienced and generous entertainer, the wedding was a social event to be long and pleasantly remembered by the friends of the bride and groom. Mrs. Banning, who for years maintained an attractive home on Fort Hill, is now living on Commonwealth Avenue. About this time, Colonel Isaac R. Dunkelberger came to Los Angeles to live, having just finished his fifth year in the army in Arizona, following a long service under Northern banners during the Civil War. While here, the Colonel met and courted Miss Mary Mallard, daughter of Judge Mallard; and on February 26th, 1867, they were married. For eight years, from March, 1877, Dunkelberger was Postmaster. He died on December 5th, 1904, survived by his widow and six children. While writing about this estimable family, it occurs to me that Mary, then a little girl, was one of the guests at my wedding. Frank Lecouvreur, who was Surveyor of Los Angeles County from 1870 until 1873, was a native of East Prussia and like his predecessor, George Hansen, came to California by way of the Horn. For a while, as I have related, he was my bookkeeper. In 1877, he married Miss Josephine Rosanna Smith who had renounced her vows as a nun. Ten years later he suffered a paralytic stroke and was an invalid until his death, on January 17th, 1901. Once introduced, the telegraph gradually grew in popularity; but even in 1870, when the Western Union company had come into the field and was operating as far as the Coast, service was anything but satisfactory. The poles between Los Angeles and San Francisco had become rotten and often fell, dragging the wires with them, and interrupting communication with the North. There were no wires, up to that time, to Santa Bárbara or San Bernardino; and only in the spring of that year was it decided to put a telegraph line through to San Diego. When the Santa Bárbara line was proposed, the citizens there speedily subscribed twenty-two hundred and forty-five dollars;
- 76. it having been the company's plan always to get some local stockholders. As the result of real estate purchases and exchanges in the late sixties and early seventies between Dr. J. S. Griffin, Phineas Banning, B. D. Wilson, P. Beaudry and others, a fruit-growing colony was planned in April, when it was proposed to take in some seventeen hundred and fifty acres of the best part of the San Pasqual rancho, including a ten-thousand-dollar ditch. A company, with a capital stock of two hundred thousand dollars divided into four thousand shares of fifty dollars each, was formed to grow oranges, lemons, grapes, olives, nuts and raisins, John Archibald being President; R. M. Widney, Vice-President; W. J. Taylor, Secretary; and the London San Francisco Bank, Treasurer. But although subscription books were opened and the scheme was advertised, nothing was done with the land until D. M. Berry and others came from Indiana and started the Indiana Colony. A rather uncommon personality for about thirty years was Fred Dohs, who came from Germany when he was twenty-three and engaged in trading horses. By 1870 he was managing a barber shop near the Downey Block, and soon after was conducting a string band. For many years, the barber-musician furnished the music for most of the local dances and entertainments, at the same time (or until prices began to be cut) maintaining his shop, where he charged two bits for a shave and four bits for a hair-cut. During his prosperity, Dohs acquired property, principally on East First Street. The first foot-bridge having finally succumbed to the turbulent waters of the erratic Los Angeles River, the great flood of 1867-68 again called the attention of our citizens to the necessity of establishing permanent and safe communication between the two sides of the stream; and this agitation resulted in the construction by Perry Woodworth of the first fairly substantial bridge at the foot of the old Aliso Road, now Macy Street, at an outlay of some twenty thousand dollars. Yet, notwithstanding the great necessity that had always existed for this improvement, it is my recollection that it was
- 77. not consummated until about 1870. Like its poor little predecessor carried away by the uncontrolled waters, the more dignified structure was broken up by a still later flood, and the pieces of timber once so carefully put together by a confident and satisfied people were strewn for a mile or two along the river banks. 'Way back in the formative years of Los Angeles, there were suddenly added to the constellation of noteworthy local characters two jovial, witty, good-for-nothing Irishmen who from the first were pals. The two were known as Dan Kelly and Micky Free. Micky's right name was Dan Harrington; but I never knew Kelly to go under any other appellation. When sober, which was not very frequent, Dan and Micky were good-natured, jocular and free from care, and it mattered not to either of them whether the morrow might find them well-fed and at liberty or in the jail then known as the Hotel de Burns: sufficient unto the day is the evil thereof was the only philosophy they knew. They were boon companions when free from drink; but when saturated, they immediately fought like demons. They were both in the toils quite ten months of the year, while during the other two months they carried a hod! Of the two, Micky was the most irredeemable, and in time he became such a nuisance that the authorities finally decided to ship him out of the country and bought him a ticket to Oregon. Micky got as far as San Pedro, where he traded his ticket for a case of delirium tremens; but he did something more—he broke his leg and was bundled back to Los Angeles, renewing here the acquaintance of both the bartender and the jailer. Some years later, he astonished the town by giving up drink and entering the Veterans's Home. When he died, they gave him a soldier's honors and a soldier's grave. In 1870, F. Bonshard imported into Los Angeles County some five or six hundred blooded Cashmere goats; and about the same time or perhaps even earlier, J. E. Pleasants conducted at Los Nietos a similar enterprise, at one time having four or five hundred of a superior breed, the wool of which brought from twenty-five to thirty-
- 78. five cents a pound. The goat-fancying Pleasants also had some twelve hundred Angoras. On June 1st, Henry Hamilton, who two years before had resumed the editorship of the Los Angeles Star, then a weekly, issued the first number of the Daily Star. He had taken into partnership George W. Barter, who three months later started the Anaheim Gazette. In 1872, Barter was cowhided by a woman, and a committee formally requested the editor to vamose the town! Barter next bought the Daily Star from Hamilton, on credit, but he was unable to carry out his contract and within a year Hamilton was again in charge. At the beginning of this decade, times in Arizona were really very bad. H. Newmark Company, who had large amounts due them from merchants in that Territory, were not entirely easy about their outstanding accounts, and this prompted Kaspare Cohn to visit our customers there. I urged him to consider the dangers of the road and to abandon his project; but he was determined to go. The story of the trip, in the light of present methods and the comparative safety of travel, is an interesting one, and I shall relate his experiences as he described them to me. He started on a Saturday, going by stage (in preference to buckboard) from Los Angeles to San Bernardino, and from there rode, as the only passenger, with a stage-driver named Brown, passing through Frink's Ranch, Gilman's, White River, Agua Caliente, Indian Wells, Toros, Dos Palmas, Chuckawalla, Mule Springs and Willow Springs. H. Newmark Company had forwarded, on a prairie schooner driven by Jesse Allen of Los Angeles, a considerable amount of merchandise which it was their intention should be sold in Arizona, and the freighting charge upon which was to be twelve and a half cents per pound. In Chuckawalla, familiarly called Chucky Valley, the travelers overtook Allen and the stock of goods; and this meeting in that lonesome region was the cause of such mutual rejoicing that Kaspare provided as abundant an entertainment as his limited stores would permit. Resuming their journey from
- 79. Chuckawalla, the driver and his companion soon left Allen and his cumbersome load in the rear. It was near Granite Wash, as they were jogging along in the evening, that they noticed some Indian fire signals. These were produced by digging a hole in the ground, filling it with combustible material, such as dry leaves, and setting fire to it. From the smoldering that resulted, smoke was emitted and sparks burst forth. Observing these ticklish warnings, the wayfarers sped away and escaped—perhaps, a tragic fate. Arriving at Ehrenberg on a Tuesday morning, Kaspare remained there all night. Still the only passenger, he left the next day; and it may be imagined how cheering, after the previous experience, was the driver's remark that, on account of the lonesome character of the trip, and especially the danger from scalping Apaches, he would never have departed without some company! Somewhere between Granite Wash and Wickenberg, a peculiar rattling revealed a near-by snake, whereupon Kaspare jumped out and shot the reptile, securing the tail and rattles. Changing horses or resting at Tyson's Wells, McMullen's and Cullen's Station, they arrived the next night at Wickenberg, the location of the Vulture Mines, where Kaspare called upon the Superintendent—a man named Peoples—to collect a large amount they owed us. Half of the sum was paid in gold bars, at the rate of sixteen dollars per ounce, while the other half we lost. A niece of M. Kremer lived in Wickenberg, where her husband was in business. She suffered a great deal from headaches, and a friend had recommended, as a talisman, the possession of snake rattles. Kaspare, with his accustomed gallantry, produced the specimen which he had obtained and gave it to the lady; and it is to be hoped that she was as permanently relieved of her pain as so many nowadays are cured of imaginary troubles by no more substantial superstitions.
- 80. Making short stops at Wilson's Station, Antelope Station, Kirkland Valley, Skull Valley and Mint Valley, Kaspare reached Prescott, some four hundred and thirty miles from San Bernardino, and enquired after Dan Hazard, the ex-Mayor's brother and one of our customers —who died about the middle of the eighties—and learned that he was then on his way to St. Louis with teams to haul back freight for Levi Bashford who, in addition to being an important trader, was Government Receiver of Public Moneys. Kaspare decided to remain in Prescott until Hazard returned; and as Jesse Allen soon arrived with the merchandise, Kaspare had ample time to sell it. Bashford, as a Government official, was not permitted to handle such goods as matches and cigars, which bore revenue stamps, but Kaspare sold him quantities of lard, beans, coffee, sugar and other supplies. He sold the revenue-stamped articles to Buffum Campbell, the former of whom had once been a well-known resident of Los Angeles. He also disposed of some goods to Henderson Brothers, afterward prominent bankers of Tucson and Globe, Arizona. In the meantime, Dan Hazard returned and settled his account in full. Kaspare remained in Prescott nearly four weeks. Between the collections that he made and the money which he received for the consigned merchandise, he had about thirteen thousand dollars in currency to bring back with him. With this amount of money on his person, the return trip was more than ever fraught with danger. Mindful of this added peril, Kaspare kept the time of his departure from Prescott secret, no one, with the exception of Bashford, being in his confidence. He prepared very quietly; and at the last moment, one Saturday afternoon, he slipped into the stage and started for California. Brown was again his companion as far as Ehrenberg. There he met Frank Ganahl and Charles Strong, both soon to become Southern Californians; and knowing them very well, their companionship contributed during the rest of the trip not only pleasure but an agreeable feeling of security. His arrival in Los Angeles afforded me much relief, and the story of his adventures and success added more than a touch of interest.
- 81. The first street-sprinklers in Los Angeles were owned and operated about the middle of July by T. W. McCracken, who was allowed by the Council to call upon residents along the route for weekly contributions to keep the water wagon going. I have told of the establishing of Hellman, Temple Company as bankers. In September, the first-named bought out his partners and continued, until 1871, as Hellman Company. With the commencement of autumn, when the belief prevailed that little or nothing could be done toward persuading the Common Council to beautify the Plaza, a movement to lay out and embellish the five-acre tract bounded by Hill and Olive, and Fifth and Sixth streets, met with such favor that, by the first week in October, some eight hundred dollars had been subscribed for the purpose. On November 19th a public meeting was held, presided over by Prudent Beaudry, Major H. M. Mitchell serving as Secretary; and it was suggested to call the proposed square the Los Angeles Park, and to enclose it, at a cost of about five hundred dollars, with a fence. Another two hundred dollars was soon made up; and the services of L. Carpenter, who offered to plow the land prior to sowing grass- seed, were accepted in lieu of a subscription. Both George Lehman and Elijah Workman showed their public spirit by planting what have since become the largest trees there. Sometime later, the name was changed to Central Park, by which it is still known. The first hackney coach ever built in Los Angeles was turned out in September by John Goller for J. J. Reynolds—about the same time that the Oriental Stage Company brought a dozen new Concord coaches from the East—and cost one thousand dollars. Goller was then famous for elaborate vehicles and patented spring buggies which he shipped even to pretentious and bustling San Francisco. Before the end of November, however, friends of the clever and enterprising carriage-maker were startled to hear that he had failed for the then not insignificant sum of about forty thousand dollars.
- 82. Up to the fall of the year, no connection existed between Temple and First Streets west of Spring; but on the first day of September, a cut through the hill, effected by means of chain-gang labor and continuing Fort Street north, was completed, to the satisfaction of the entire community. About the middle of October, a petition was presented to the Common Council calling attention to the fact that the Los Angeles Water Company two years before had agreed to erect a fountain on the Plaza; and declaring that the open place was little short of a scarecrow for visitors. The Company immediately replied that it was ready to put up the fountain; and in November the Council ordered the brick tank taken away. At the beginning of August, 1871, the fountain began playing. During the second marshalship of William C. Warren, when Joe Dye was one of his deputy officers, there was great traffic in Chinese women, one of whom was kidnaped and carried off to San Diego. A reward of a hundred dollars was offered for her return, and she was brought back on a charge of theft and tried in the Court of Justice Trafford, on Temple Street near Spring. During the trial, on October 31st, 1870, Warren and Dye fell into a dispute as to the reward; and the quarrel was renewed outside the courtroom. At a spot near the corner of Spring and Temple streets Dye shot and killed Warren; and in the scrimmage several other persons standing near were wounded. Dye was tried, but acquitted. Later, however, he himself was killed by a nephew, Mason Bradfield, whose life he had frequently threatened and who fired the deadly bullet from a window of the New Arlington Hotel, formerly the White House, at the southeast corner of Commercial and Los Angeles streets. Mrs. C. P. Bradfield, Bradfield's mother and a teacher, who came in 1875, was the author of certain text-books for drawing, published by A. S. Barnes Company of New York. Failures in raising and using camels in the Southwest were due, at least partially, to ignorance of the animal's wants, a company of Mexicans, in the early sixties, overloading some and treating them
- 83. so badly that nearly all died. Later, Frenchmen, who had had more experience, secured the two camels left, and by 1870 there was a herd of no less than twenty-five on a ranch near the Carson River in Nevada, where they were used in packing salt for sixty miles or more to the mills. On October 31st, the first Teacher's Institute held in Los Angeles County was opened, with an attendance of thirty-five, in the old Bath Street schoolhouse, that center being selected because the school building at Spring and Second streets, though much better adapted to the purpose, was considered to be too far out of town! County Superintendent W. M. McFadden was President; J. M. Guinn was Vice-President; and P. C. Tonner was Secretary; while a leader in discussions was Dr. Truman H. Rose, who there gave a strong impetus to the founding of the first high school. Soon after this Institute was held, the State Legislature authorized bonds to the amount of twenty thousand dollars for the purpose of erecting another schoolhouse; and the building was soon to be known as the Los Angeles High School. W. H. Workman, M. Kremer and H. D. Barrows were the building committee. Mentioning educators, I may introduce the once well-known name of Professor Adams, an instructor in French who lived here in the early seventies. He was so very urbane that on one occasion, while overdoing his polite attention to a lady, he fell off the sidewalk and badly broke his leg! In a previous chapter I have spoken of a Frenchman named Lachenais who killed a fellow-countryman at a wake, the murder being one of a succession of crimes for which he finally paid the penalty at the hands of a Vigilance Committee in the last lynching witnessed here. Lachenais lived near where the Westminster Hotel now stands, on the northeast corner of Main and Fourth streets, but he also had a farm south of the city, adjoining that of Jacob Bell who was once a
- 84. partner in sheep-raising with John Schumacher. The old man was respectable and quiet, but Lachenais quarreled with him over water taken from the zanja. Without warning, he rode up to Bell as he was working in his field and shot him dead; but there being no witnesses to the act, this murder remained, temporarily, a mystery. One evening, as Lachenais (to whom suspicion had been gradually directed), was lounging about in a drunken condition, he let slip a remark as to the folly of anyone looking for Bell's murderer; and this indiscretion led to his arrest and incarceration. No sooner had the news of Lachenais's apprehension been passed along than the whole town was in a turmoil. A meeting at Stearns's Hall was largely attended; a Vigilance Committee was formed; Lachenais's record was reviewed and his death at the hands of an outraged community was decided upon. Everything being arranged, three hundred or more armed men, under the leadership of Felix Signoret, the barber—Councilman in 1863 and proprietor of the Signoret Building opposite the Pico House—assembled on the morning of December 17th, marched to the jail, overcame Sheriff Burns and his assistants, took Lachenais out, dragged him along to the corral of Tomlinson Griffith (at the corner of Temple and New High streets) and there summarily hanged him. Then the mob, without further demonstration, broke up; the participants going their several ways. The reader may have already observed that this was not the first time that the old Tomlinson Griffith gate had served this same gruesome purpose. The following January, County Judge Y. Sepúlveda charged the Grand Jury to do its duty toward ferreting out the leaders of the mob, and so wipe out this reproach to the city; but the Grand Jury expressed the conviction that if the law had hitherto been faithfully executed in Los Angeles, such scenes in broad daylight would never have taken place. The editor of the News, however, ventured to assert that this report was but another disgrace.
- 85. CHAPTER XXIX THE CHINESE MASSACRE 1871 H. Newmark Company enjoyed associations with nearly all of the most important wool men and rancheros in Southern California, our office for many years being headquarters for these stalwarts, as many as a dozen or more of whom would ofttimes congregate, giving the store the appearance of a social center. They came in from their ranches and discussed with freedom the different phases of their affairs and other subjects of interest. Wheat, corn, barley, hay, cattle, sheep, irrigation and kindred topics were passed upon; although in 1871 the price of wool being out of all proportion to anything like its legitimate value, the uppermost topic of conversation was wool. These meetings were a welcome interruption to the monotony of our work. Some of the most important of these visitors were Jotham, John W. and Llewellyn Bixby, Isaac Lankershim, L. J. Rose, I. N. Van Nuys, R. S. Baker, George Carson, Manuel Dominguez, Domingo Amestoy, Juan Matías Sanchez, Dan Freeman, John Rowland, John Reed, Joe Bridger, Louis Phillips, the brothers Garnier, Remi Nadeau, E. J. Baldwin, P. Banning and Alessandro Repetto. There was also not a weather prophet, near or far, who did not manage to appear at these weighty discussions and offer his oracular opinions about the pranks of the elements; on
- 86. which occasions, one after another of these wise men would step to the door, look at the sky and broad landscape, solemnly shake his head and then render his verdict to the speculating circle within. According as the moon emerged so that one could hang something upon it, or in such a manner that water would run off (as they pictured it), we were to have dry or rainy weather; nor would volumes of talk shake their confidence. Occasionally, I added a word, merely to draw out these weather-beaten and interesting old chaps; but usually I listened quietly and was entertained by all that was said. Hours would be spent by these friends in chatting and smoking the time away; and if they enjoyed the situation half as much as I did, pleasant remembrances of these occasions must have endured with them. Many of those to whom I have referred have ended their earthly careers, while others, living in different parts of the county, are still hale and hearty. A curious character was then here, in the person of the reputed son of a former, and brother of the then, Lord Clanmorris, an English nobleman. Once a student at Dr. Arnold's famous Rugby, he had knocked about the world until, shabbily treated by Dame Fortune, he had become a sheepherder in the employ of the Bixbys. M. J. Newmark, who now came to visit us from New York, was admitted to partnership with H. Newmark Company, and this determined his future residence. As was natural in a town of pueblo origin, plays were often advertised in Spanish; one of the placards, still preserved, thus announcing the attraction for January 30th, at the Merced Theater:
- 87. TEATRO MERCED LOS ANGELES Lunes, Enero 30, de 1871 Primero Función de la Gran Compañia Dramática, De Don Tomás Maguire, El Empresario Veterano de San Francisco, Veinte y Cuatro Artistas de ambos sexos, todos conocidos como Estrellas de primera clase. In certain quarters of the city, the bill was printed in English. Credit for the first move toward the formation of a County Medical Society here should probably be given to Dr. H. S. Orme, at whose office early in 1871 a preliminary meeting was held; but it was in the office of Drs. Griffin and Widney, on January 31st, that the organization was effected, my friend Griffin being elected President; Dr. R. T. Hayes, Vice-President; Dr. Orme, Treasurer; and Dr. E. L. Dow, Secretary. Thus began a society which, in the intervening years, has accomplished much good work. Late in January, Luther H. Titus, one of several breeders of fast horses, brought from San Francisco by steamer a fine thoroughbred stallion named Echo, a half-brother of the celebrated trotter Dexter which had been shipped from the East in a Central Pacific car especially constructed for the purpose—in itself something of a wonder then. Sporting men came from a distance to see the horse; but interest was divided between the stallion and a mammoth turkey of a peculiar breed, also brought west by Titus, who prophesied that the bird, when full grown, would tip the beam at from forty-five to fifty pounds. Early in February, the first steps were taken to reorganize and consolidate the two banking houses in which Downey and Hellman were interested, when it was proposed to start the Bank of Los Angeles, with a capital of five hundred thousand dollars. Some three hundred and eighty thousand dollars of this sum were soon subscribed; and by the first week in April, twenty-five per cent. of the capital had been called in. John G. Downey was President and I.
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