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© 2015

Topological Dimension and Dynamical Systems

Benefits

  • Provides an accessible introduction to difficult material previously only found scattered in literature

  • Provides a detailed exposition of the elements of dimension theory for topological spaces and dynamical systems

  • Contains 160 exercises, of which helpful hints are provided for the most challenging

Textbook

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xv
  2. Topological Dimension

    1. Front Matter
      Pages 1-1
    2. Michel Coornaert
      Pages 3-25
    3. Michel Coornaert
      Pages 27-48
    4. Michel Coornaert
      Pages 49-67
    5. Michel Coornaert
      Pages 69-86
    6. Michel Coornaert
      Pages 87-104
  3. Mean Topological Dimension

    1. Front Matter
      Pages 105-105
    2. Michel Coornaert
      Pages 107-121
    3. Michel Coornaert
      Pages 123-138
    4. Michel Coornaert
      Pages 157-190
  4. Back Matter
    Pages 223-233

About this book

Introduction

Translated from the popular French edition, the goal of the book is to provide a self-contained introduction to mean topological dimension, an invariant of dynamical systems introduced in 1999 by Misha Gromov. The book examines how this invariant was successfully used by Elon Lindenstrauss and Benjamin Weiss to answer a long-standing open question about embeddings of minimal dynamical systems into shifts.

A large number of revisions and additions have been made to the original text. Chapter 5 contains an entirely new section devoted to the Sorgenfrey line. Two chapters have also been added: Chapter 9 on amenable groups and Chapter 10 on mean topological dimension for continuous actions of countable amenable groups. These new chapters contain material that have never before appeared in textbook form. The chapter on amenable groups is based on Følner’s characterization of amenability and may be read independently from the rest of the book.

Although the contents of this book lead directly to several active areas of current research in mathematics and mathematical physics, the prerequisites needed for reading it remain modest; essentially some familiarities with undergraduate point-set topology and, in order to access the final two chapters, some acquaintance with basic notions in group theory. Topological Dimension and Dynamical Systems is intended for graduate students, as well as researchers interested in topology and dynamical systems. Some of the topics treated in the book directly lead to research areas that remain to be explored.

Keywords

Amenable Group Benjamin Weiss Covering Dimension Dynamical Systems Elon Lindenstrauss Mean Topological Dimension Misha Gromov Shift Subshift Symbolic Dynamics Topological Invariant

Authors and affiliations

  1. 1.Institut de Recherche Mathématique AvancéeUniversity of StrasbourgStrasbourgFrance

About the authors

Professor Michel Coornaert teaches at the University of Strasbourg. He is the author of many Springer volumes including Cellular Automata and Groups (2010), Symbolic Dynamics and Hyperbolic Groups (1993) and Geometrie et theorie des groupes (1990).

Bibliographic information

  • Book Title Topological Dimension and Dynamical Systems
  • Authors Michel Coornaert
  • Series Title Universitext
  • Series Abbreviated Title Universitext
  • DOI https://doi.org/10.1007/978-3-319-19794-4
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-319-19793-7
  • eBook ISBN 978-3-319-19794-4
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages XV, 233
  • Number of Illustrations 12 b/w illustrations, 1 illustrations in colour
  • Additional Information Original French edition published by Société Mathématique de France, Paris, 2005
  • Topics Dynamical Systems and Ergodic Theory
    Topology
  • Buy this book on publisher's site
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