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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.
List of contents
Part I. Universal Constructions: 1. The Stone-Cech compactification ßT; Appendix to Chapter 1. Ultrafilters and the construction of ßT; 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.
About the author
David B. Ellis received his PhD in algebraic topology from the University of California, Berkeley. He has taught at a wide variety of institutions including Yale University, Vassar College, and Washington University in St Louis, and is currently a member of the faculty at Beloit College in Wisconsin. He has published papers in algebraic topology, foliations, fractal geometry and topological dynamics.
Summary
An original and self-contained treatment of some key ideas in the abstract theory of topological dynamics, focussing on the role of automorphisms and equivalence relations. This approach leads to simpler proofs of classical results and provides motivation for further study, making this lucid book ideal for graduate students and researchers.
