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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 Folner'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.
Gromov's theory of hyperbolic groups have had a big impact in
combinatorial group theory and has deep connections with many
branches of mathematics suchdifferential geometry, representation
theory, ergodic theory and dynamical systems. This book is an
elaboration on some ideas of Gromov on hyperbolic spaces and
hyperbolic groups in relation with symbolic dynamics. Particular
attention is paid to the dynamical system defined by the action of
a hyperbolic group on its boundary. The boundary is most
oftenchaotic both as a topological space and as a dynamical system,
and a description of this boundary and the action is given in terms
of subshifts of finite type. The book is self-contained and
includes two introductory chapters, one on Gromov's hyperbolic
geometry and the other one on symbolic dynamics. It is intended for
students and researchers in geometry and in dynamical systems, and
can be used asthe basis for a graduate course on these subjects.
Cellular automata were introduced in the first half of the last
century by John von Neumann who used them as theoretical models for
self-reproducing machines. The authors present a self-contained
exposition of the theory of cellular automata on groups and explore
its deep connections with recent developments in geometric group
theory, symbolic dynamics, and other branches of mathematics and
theoretical computer science. The topics treated include in
particular the Garden of Eden theorem for amenable groups, and the
Gromov-Weiss surjunctivity theorem as well as the solution of the
Kaplansky conjecture on the stable finiteness of group rings for
sofic groups. The volume is entirely self-contained, with 10
appendices and more than 300 exercises, and appeals to a large
audience including specialists as well as newcomers in the field.
It provides a comprehensive account of recent progress in the
theory of cellular automata based on the interplay between
amenability, geometric and combinatorial group theory, symbolic
dynamics and the algebraic theory of group rings which are treated
here for the first time in book form.
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