PART I Ergodicity and Mixing. Examples of Dynamical Systems
1. Basic Definitions of Ergodic Theory.
2. Smooth Dynamical Systems on Smooth Manifolds
3. Smooth Dynamical Systems on the Torus
4. Dynamical Systems of Algebraic Origin
5. Interval Exchange Transformations
6. Billiards
7. Dynamical Systems in Number Theory
8. Dynamical Systems in Probability Theory
9. Examples of Infinite Dimensional Dynamical Systems

PART II Basic Constructions of Ergodic Theory
10. Simplest General Constructions and Elements of Entropy Theory of Dynamical Systems
11. Special Representations of Flows

PART III Spectral Theory of Dynamical Systems
12. Dynamical Systems with Pure Point Spectrum
13. Examples of Spectral Analysis of Dynamical Systems
14. Spectral Analysis of Gauss Dynamical Systems

PART IV Approximation Theory of Dynamical Systems by Periodic Dynamical Systems and Some of its Applications
15. Approximations of Dynamical Systems
16. Special Representations and Approximations of Smooth Dynamical Systems on the Two-dimensional Torus

Ergodic theory is one of the few branches of mathematics which has changed radically during the last two decades. Before this period, with a small number of exceptions, ergodic theory dealt primarily with averaging problems and general qualitative questions, while now it is a powerful amalgam of methods used for the analysis of statistical properties of dyna­ mical systems. For this reason, the problems of ergodic theory now interest not only the mathematician, but also the research worker in physics, biology, chemistry, etc. The outline of this book became clear to us nearly ten years ago but, for various reasons, its writing demanded a long period of time. The main principle, which we adhered to from the beginning, was to develop the approaches and methods or ergodic theory in the study of numerous concrete examples.