Ergodic Theory: (Ergodic Theory)

This volume in the Encyclopedia of Complexity and Systems Science, Second Edition, covers recent developments in classical areas of ergodic theory, including the asymptotic properties of measurable dynamical systems, spectral theory, entropy, ergodic theorems, joinings, isomorphism theory, recurrence, nonsingular systems. It enlightens connections of ergodic theory with symbolic dynamics, topological dynamics, smooth dynamics, combinatorics, number theory, pressure and equilibrium states, fractal geometry, chaos. In addition, the new edition includes dynamical systems of probabilistic origin, ergodic aspects of Sarnak's conjecture, translation flows on translation surfaces, complexity and classification of measurable systems, operator approach to asymptotic properties, interplay with operator algebras

Table of Contents:
Introduction to Ergodic Theory.- Ergodic Theory: Basic Examples and Constructions.- Ergodicity and Mixing Properties.- Ergodic Theory: Recurrence.- Ergodic Theorems.- Spectral Theory of Dynamical Systems.- Joinings in Ergodic Theory.- Entropy in Ergodic Theory.- Isomorphism Theory in Ergodic Theory.- Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems.- Ergodic Theory: Non-singular Transformations.- Sarnak’s Conjecture from the Ergodic Theory Point of View.- Smooth Ergodic Theory.- Ergodic and spectral theory of area-preserving flows on surfaces.- Pressure and Equilibrium States in Ergodic Theory.- Parallels Between Topological Dynamics and Ergodic Theory.- Symbolic Dynamics.- Operator ergodic theory.- Dynamical Systems and C-algebras.- The complexity and the structure and classification of Dynamical Systems.- Ergodic Theory: Interactions with Combinatorics and Number Theory.- Ergodic Theory on Homogeneous Spaces and Metric Number Theory.- Ergodic Theory:Rigidity.- Chaos and Ergodic Theory.- Ergodic Theory: Fractal Geometry


About the Author :

Cesar E. Silva is the Hagey Family Professor of Mathematics at Williams College, where he has taught since 1984, and was chair of the department in 2008-2009 and 2010-2012. He received his PhD from the University of Rochester under the supervision of Dorothy Maharam, and his BS from the Pontificia Universidad Catolica in Peru. He has held visiting positions at a number of universities including Maryland and Toronto.  In 2015, Silva was elected a Fellow of the American Mathematical Society. Silva’s research interests are in ergodic theory and dynamical systems, in particular the dynamics of nonsingular and infinite measure-preserving transformations, rank-one transformations, and p-adic dynamics. Silva is the author of Invitation to Ergodic Theory and Invitation to Real Analysis, both published by the American Mathematical Society, and is co-editor of two volumes of conference proceedings, published in the AMS Contemporary Mathematics series. He was associate editor of AMS Notices and is the author or co-author of over 50 research articles in the general area of ergodic theory.

“Together the articles give a useful overview of the key ideas in ergodic theory across a great diversity of settings. As a reference to precise formulations of a huge array of concepts and methods in ergodic theory together with extensive specialised bibliographies in each article, this collection will undoubtedly be useful to researchers and postgraduate students in any field that has reason to call on these ideas.” (Thomas B. Ward, zbMATH 1532.37003, 2024)