Topological Dimension and Dynamical Systems

✍ Scribed by Michel Coornaert (auth.)

• Publisher: Springer International Publishing
• Year: 2015
• Tongue: English
• Leaves: 239
• Series: Universitext
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages i-xv
Front Matter....Pages 1-1
Topological Dimension....Pages 3-25
Zero-Dimensional Spaces....Pages 27-48
Topological Dimension of Polyhedra....Pages 49-67
Dimension and Maps....Pages 69-86
Some Classical Counterexamples....Pages 87-104
Front Matter....Pages 105-105
Mean Topological Dimension for Continuous Maps....Pages 107-121
Shifts and Subshifts over (\mathbb {Z}) ....Pages 123-138
Applications of Mean Dimension to Embedding Problems....Pages 139-155
Amenable Groups....Pages 157-190
Mean Topological Dimension for Actions of Amenable Groups....Pages 191-222
Back Matter....Pages 223-233

✦ Subjects

Dynamical Systems and Ergodic Theory; Topology