Ergodic Theory and Dynamical Systems

Description

This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics. This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors. Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader. Front Matter....Pages i-xiii Front Matter....Pages 1-1 The Mean Ergodic Theorem....Pages 3-14 The Pointwise Ergodic Theorem....Pages 15-24 Mixing....Pages 25-33 The Hopf Argument....Pages 35-46 Front Matter....Pages 47-47 Topological Dynamics....Pages 49-57 Nonwandering....Pages 59-67 Conjugation....Pages 69-78 Linearization....Pages 79-88 A Strange Attractor....Pages 89-98 Front Matter....Pages 99-99 Entropy....Pages 101-112 Entropy and Information Theory....Pages 113-121 Computing Entropy....Pages 123-132 Front Matter....Pages 133-133 Lebesgue Spaces and Isomorphisms....Pages 135-144 Ergodic Decomposition....Pages 145-154 Measurable Partitions and σ-Algebras....Pages 155-163 Front Matter....Pages 165-165 Weak Convergence....Pages 167-170 Conditional Expectation....Pages 171-174 Topology and Measures....Pages 175-180 Back Matter....Pages 181-190