Ergodic Theory

1. Front Matter

   Pages i-x

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2. Ergodicity and Mixing. Examples of Dynamical Systems

   1. Front Matter

      Pages 1-1

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   2. Basic Definitions of Ergodic Theory

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 3-42

   3. Smooth Dynamical Systems on Smooth Manifolds

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 43-63

   4. Smooth Dynamical Systems on the Torus

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 64-95

   5. Dynamical Systems of Algebraic Origin

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 96-121

   6. Interval Exchange Transformations

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 122-137

   7. Billiards

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 138-156

   8. Dynamical Systems in Number Theory

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 157-177

   9. Dynamical Systems in Probability Theory

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 178-192

   10. Examples of Infinite Dimensional Dynamical Systems

       • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

       Pages 193-224

3. Basic Constructions of Ergodic Theory

   1. Front Matter

      Pages 225-225

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   2. Simplest General Constructions and Elements of Entropy Theory of Dynamical Systems

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 227-291

   3. Special Representations of Flows

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 292-321

4. Spectral Theory of Dynamical Systems

   1. Front Matter

      Pages 323-323

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   2. Dynamical Systems with Pure Point Spectrum

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 325-337

   3. Examples of Spectral Analysis of Dynamical Systems

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 338-355

   4. Spectral Analysis of Gauss Dynamical Systems

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 356-385

5. Approximation Theory of Dynamical Systems by Periodic Dynamical Systems and Some of its Applications

   1. Front Matter

      Pages 387-387

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   2. Approximations of Dynamical Systems

      • I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

      Pages 389-407

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. Because of this, Part I of the book contains the description of various classes of dynamical systems, and their elementary analysis on the basis of the fundamental notions of ergodicity, mixing, and spectra of dynamical systems. Here, as in many other cases, the adjective" elementary" i~ not synonymous with "simple. " Part II is devoted to "abstract ergodic theory. " It includes the construc­ tion of direct and skew products of dynamical systems, the Rohlin-Halmos lemma, and the theory of special representations of dynamical systems with continuous time. A considerable part deals with entropy.

• Elementary Analysis
• Ergodentheorie
• Ergodic theory
• ergodicity
• mixing

• Landau Institute of Theoretical Physics, Academy of Sciences, Moscow, USSR

  I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

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