Following the concept of the EMS series this volume sets out to familiarize the reader to the fundamental ideas and results of modern ergodic theory and to its applications to dynamical systems and statistical mechanics. The exposition starts from the basic of the subject, introducing ergodicity, mixing and entropy. Then the ergodic theory of smooth dynamical systems is presented - hyperbolic theory, billiards, one-dimensional systems and the elements of KAM theory. Numerous examples are presented carefully along with the ideas underlying the most important results. The last part of the book deals with the dynamical systems of statistical mechanics, and in particular with various kinetic equations. This book is compulsory reading for all mathematicians working in this field, or wanting to learn about it.

Inhalt

I. General Ergodic Theory of Groups of Measure Preserving Transformations.- 1. Basic Notions of Ergodic Theory and Examples of Dynamical Systems.- 2. Spectral Theory of Dynamical Systems.- 3. Entropy Theory of Dynamical Systems.- 4. Periodic Approximations and Their Applications. Ergodic Theorems, Spectral and Entropy Theory for the General Group Actions.- 5. Trajectory Theory.- II. Ergodic Theory of Smooth Dynamical Systems.- 6. Stochasticity of Smooth Dynamical Systems. The Elements of KAM-Theory.- 7. General Theory of Smooth Hyperbolic Dynamical Systems.- 8. Dynamical Systems of Hyperbolic Type with Singularities.- 9. Ergodic Theory of One-Dimensional Mappings.- III. Dynamical Systems of Statistical Mechanics and Kinetic Equations.- 10. Dynamical Systems of Statistical Mechanics.- 11. Existence and Uniqueness Theorems for the Boltzmann Equation.