Automorphisms and Equivalence Relations in Topological Dynamics

A lucid and self-contained treatment of many key ideas in topological dynamics, achieved by focusing on equivalence relations and automorphisms.

David B. Ellis (Author), Robert Ellis (Author)

9781107633223, Cambridge University Press

Paperback / softback, published 5 June 2014

281 pages, 80 exercises
22.9 x 15.2 x 1.6 cm, 0.42 kg

Focusing on the role that automorphisms and equivalence relations play in the algebraic theory of minimal sets provides an original treatment of some key aspects of abstract topological dynamics. Such an approach is presented in this lucid and self-contained book, leading to simpler proofs of classical results, as well as providing motivation for further study. Minimal flows on compact Hausdorff spaces are studied as icers on the universal minimal flow M. The group of the icer representing a minimal flow is defined as a subgroup of the automorphism group G of M, and icers are constructed explicitly as relative products using subgroups of G. Many classical results are then obtained by examining the structure of the icers on M, including a proof of the Furstenberg structure theorem for distal extensions. This book is designed as both a guide for graduate students, and a source of interesting new ideas for researchers.

Part I. Universal Constructions: 1. The Stone–Cech compactification ?T
Appendix to Chapter 1. Ultrafilters and the construction of ?T
2. Flows and their enveloping semigroups
3. Minimal sets and minimal right ideals
4. Fundamental notions
5. Quasi-factors and the circle operator
Appendix to Chapter 5. The Vietoris topology on 2^X
Part II. Equivalence Relations and Automorphisms: 6. Quotient spaces and relative products
7. Icers on M and automorphisms of M
8. Regular flows
9. The quasi-relative product
Part III. The ?-Topology: 10. The ?-topology on Aut(X)
11. The derived group
12. Quasi-factors and the ?-topology
Part IV. Subgroups of G and the Dynamics of Minimal Flows: 13. The proximal relation and the group P
14. Distal flows and the group D
15. Equicontinuous flows and the group E
Appendix to Chapter 15. Equicontinuity and the enveloping semigroup
16. The regionally proximal relation
Part V. Extensions of Minimal Flows: 17. Open and highly proximal extensions
Appendix. Extremely disconnected flows
18. Distal extensions of minimal flows
19. Almost periodic extensions
20. A tale of four theorems.

Subject Areas: Topology [PBP], Mathematics [PB]