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Automorphisms and Equivalence Relations in Topological Dynamics

$72.99 (C)

Part of London Mathematical Society Lecture Note Series

  • Date Published: August 2014
  • availability: In stock
  • format: Paperback
  • isbn: 9781107633223

$ 72.99 (C)
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About the Authors
  • 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.

    • The authors' original approach provides a clearer and simpler treatment of some key ideas and classical results
    • Provides plenty of scope for further research
    • The self-contained exposition and detailed proofs give a level of rigour that will appeal to both novices and experts
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    Product details

    • Date Published: August 2014
    • format: Paperback
    • isbn: 9781107633223
    • length: 281 pages
    • dimensions: 229 x 152 x 16 mm
    • weight: 0.42kg
    • contains: 80 exercises
    • availability: In stock
  • Table of Contents

    Part I. Universal Constructions:
    1. The Stone–Cech compactification βT
    Appendix to Chapter 1. Ultrafilters and the construction of βT
    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 τ-Topology:
    10. The τ-topology on Aut(X)
    11. The derived group
    12. Quasi-factors and the τ-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.

  • Authors

    David B. Ellis, Beloit College, Wisconsin
    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.

    Robert Ellis, University of Minnesota
    Robert Ellis, a student of W. Gottschalk, is one of the founders of topological dynamics. He received his PhD from the University of Pennsylvania. During his long and productive career he made many major contributions to the abstract theory of topological dynamics, including his joint continuity theorem and the introduction of the enveloping semigroup. In so doing he laid the foundation for an algebraic approach to topological dynamics. Professor Ellis retired from the University of Minnesota in 1995 and was named a fellow of the AMS in 2012.

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