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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 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.

Mathematics. --- Dynamical Systems and Ergodic Theory. --- Topology. --- Differentiable dynamical systems. --- Mathématiques --- Dynamique différentiable --- Topologie --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Dynamics. --- Ergodic theory. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Topological dynamics. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics

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Group theory --- Differential geometry. Global analysis --- Differentiable dynamical systems --- Differentieerbare dynamicasystemen --- Dynamique topologique --- Global differential geometry --- Groupes hyperboliques --- Géométrie différentielle globale --- Hyperbolic groups --- Hyperbolische groepen --- Meetkunde [Differentiaal globale ] --- Systèmes dynamiques différentiables --- Topological dynamics --- Topologische dynamica --- Global differential geometry. --- Hyperbolic groups. --- Differentiable dynamical systems. --- 51 --- Dynamics, Topological --- Geometry, Differential --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Mathematics --- 51 Mathematics --- Topological dynamics.

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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.

Cellular automata. --- Computational complexity. --- Electronic books. -- local. --- Cellular automata --- Group theory --- Computational complexity --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Calculus --- Computers, Iterative circuit --- Iterative circuit computers --- Structures, Tessellation (Automata) --- Tessellation structures (Automata) --- Complexity, Computational --- Computer science. --- Computer logic. --- Algebra. --- Dynamics. --- Ergodic theory. --- Computer Science. --- Logics and Meanings of Programs. --- Dynamical Systems and Ergodic Theory. --- Ergodic transformations --- Continuous groups --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Mathematical analysis --- Computer science logic --- Logic, Symbolic and mathematical --- Informatics --- Science --- Parallel processing (Electronic computers) --- Pattern recognition systems --- Sequential machine theory --- Electronic data processing --- Machine theory --- Logic design. --- Differentiable dynamical systems. --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Design, Logic --- Design of logic systems --- Digital electronics --- Electronic circuit design --- Logic circuits --- Switching theory --- Group theory.

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This unique book provides a self-contained exposition of the theory of cellular automata on groups and explores its deep connections with recent developments in geometric and combinatorial group theory, amenability, symbolic dynamics, the algebraic theory of group rings, and other branches of mathematics and theoretical computer science. The topics treated include the Garden of Eden theorem for amenable groups, the Gromov–Weiss surjunctivity theorem, and the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups. Entirely self-contained and now in its second edition, the volume includes 10 appendices and more than 600 exercises, the solutions of which are presented in the companion book Exercises in Cellular Automata and Groups (2023) by the same authors. It will appeal to a large audience, including specialists and newcomers to the field.

Algebra. --- Dynamical systems. --- Computer science. --- Dynamical Systems. --- Computer Science Logic and Foundations of Programming. --- Autòmats cel·lulars

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