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.
Yves Coudene is a full professor at Brest University, France. His research areas include hyperbolic dynamics, ergodic theory and the geometry of negatively curved spaces.
Introduction.- Part I Ergodic Theory.- The Mean Ergodic Theorem.- The Pointwise Ergodic Theorem.- Mixing.- The Hopf Argument.- Part II Dynamical Systems.- Topological Dynamics.- Nonwandering.- Conjugation.- Linearization.- A Strange Attractor.- Part III Entropy Theory.- Entropy.- Entropy and Information Theory.- Computing Entropy.- Part IV Ergodic Decomposition.- Lebesgue Spaces and Isomorphisms.- Ergodic Decomposition.- Measurable Partitions and -Algebras.- Part V Appendices.- Weak Convergence.- Conditional Expectation.- Topology and Measures.- References.