main-content

Über dieses Buch

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.

Inhaltsverzeichnis

Chapter 1. Basic Definitions of Ergodic Theory

Abstract
Ergodic theory studies motion in a measure space. Therefore we begin by considering the notion of measure space.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 2. Smooth Dynamical Systems on Smooth Manifolds

Abstract
One of the most important classes of dynamical systems are those which are determined by differentiable maps of smooth manifolds. As a rule, by a manifold we shall mean an m-dimensional compact closed orientable manifold of class C (m ≥ 1).
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 3. Smooth Dynamical Systems on the Torus

Abstract
Diffeomorphisms and flows on tori are of particular importance from various points of view. It might at first seem that this is a very special class of dynamical systems. However, this is not so: many important dynamical systems turn out to be nonergodic and their phase spaces split into invariant tori (see §3, Chap. 2).
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 4. Dynamical Systems of Algebraic Origin

Abstract
In this chapter we shall consider certain classes of dynamical systems of algebraic origin. In these systems the phase space possesses some sort of symmetry, and the action of the dynamical system preserves this symmetry. We shall make use of the theory of characters of commutative compact groups. This theory is developed, for example, in the book by Pontrjagin [1].
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 5. Interval Exchange Transformations

Abstract
Suppose the space M is the semi-interval [0,1), ξ = (Δ1,..., Δ r ) is a partition of M into r ≥ 2 disjoint semi-intervals, numbered from left to right, and let π = (π 1,..., π r ) be a permutation of the number (1, 2,..., r).
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 6. Billiards

Abstract
In this chapter we consider dynamical systems of the billiards type, i.e., dynamical systems corresponding to the inertial motion of a point mass inside a domain with a piece-wise smooth boundary. Upon reaching the boundary, the point bounces off in accordance to the usual rule: “the angle of incidence is equal to the angle of reflection.” Besides the intrinsic interest of the problem, systems of billiards are remarkable in view of the fact that they naturally appear in many important problems of physics.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 7. Dynamical Systems in Number Theory

Abstract
Many problems in number theory may be stated as problems relating to the uniform distribution of certain numerical sequences. Recall that the sequence x 1, x 2,...., 0 ≤ x n ≤ 1, is uniformly distributed on the closed interval [0, 1], if for any function fC([0, 1]) we have the relation
$$\mathop{{\lim }}\limits_{{n \to \infty }} \frac{1}{n}\sum\limits_{{k = 1}}^{n} {f({x_{k}}) = \int_{0}^{1} {f(x)dx} }$$
. Similarly, we can define the uniform distribution on an arbitrary closed interval [a, b], a < b.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 8. Dynamical Systems in Probability Theory

Abstract
Suppose M is the set of all sequences, infinite in both directions x = (..., y -1, y 0, y 1,...), whose coordinates y i are points of a fixed measurable space (Y, 𝔄). M possesses a natural σ-algebra 𝔖̃ generated by cylindrical sets, i.e., sets of the form
$$A = \{ x = (...,{y_{{ - 1}}},{y_{0}},{y_{1}},...) \in M:{y_{{{i_{1}}}}} \in {C_{1}},...,{y_{{{i_{r}}}}} \in {C_{r}}\} ,$$
(1)
where 1 ≤ r < ∞, i 1,..., i r are integers and C 1,..., C r ∈ 𝔄. Suppose μ is a normalized measure on 𝔖̃ and 𝔖 is the completion of 𝔖̃ with respect to the measure μ. In probability theory the triple (M, 𝔖, μ) is said to be a discrete time random process and the space (Y, 𝔄) is the state space of this process.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 9. Examples of Infinite Dimensional Dynamical Systems

Abstract
In this section we consider one of the simplest examples of infinite-dimensional dynamical systems—an ideal gas consisting of an infinite number of noninteracting particles. We begin with the case corresponding to the motion of particles in Euclidian space ℝ d , d ≥ 1.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 10. Simplest General Constructions and Elements of Entropy Theory of Dynamical Systems

Abstract
Consider the measure space (M, 𝔖, μ) which is the Cartesian product of two other measure spaces
$$(M,,\mu ) = ({M_1} \times {M_2},{_1} \times {_2},{\mu _1} \times {\mu _2})$$
Assume that the dynamical systems {T 1 t }, {T 2 t } act in the factors, either both with discrete time or both with continuous time.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 11. Special Representations of Flows

Abstract
There is a general method in ergodic theory which reduces many problems concerning dynamical systems with continuous time to the corresponding problem for dynamical systems with discrete time. This method goes back to Poincaré; for the study of trajectories of a smooth dynamical system in the neighborhood of a closed trajectory he proposed to consider the “return” map which arises on a transversal surface of codimension 1 to the closed trajectory: the transformation consists in following the trajectory starting at a given point of the surface until its next intersection with the surface.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 12. Dynamical Systems with Pure Point Spectrum

Abstract
In this chapter we study an important class of dynamical systems—dynamical systems with pure point spectrum. Concerning the notions of the spectral theory of unitary operators used here see Appendix 2.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 13. Examples of Spectral Analysis of Dynamical Systems

Abstract
By the spectrum of a dynamical system we mean the spectrum of a unitary operator or group (semigroup) of unitary operators adjoint to the system on the invariant subset L 0 2 (M, 𝔖, μ) of functions of zero mean. Two dynamical systems with the same spectrum are said to be spectrally equivalent. In this chapter we shall compute the spectrum of certain dynamical systems. The necessary facts from the spectral theory of unitary operators are provided in Appendix 2.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 14. Spectral Analysis of Gauss Dynamical Systems

Abstract
Gauss dynamical systems were introduced in §2 of Chap. 8. There we constructed the real and complex subspaces H 1 (r) , H 1 (c) of Hilbert space L 2(M, 𝔖, μ), where μ is the Gauss measure on the space M and, using them, obtained a necessary condition for the ergodicity of a Gauss dynamical system, consisting in the continuity of the spectral measure σ corresponding to the measure μ.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 15. Approximations of Dynamical Systems

Abstract
The constructive theory of functions studies the relationship between the properties of functions and the speed of their approximations by functions of some particular fixed class. In a similar way, in ergodic theory we may study the dependence of various properties of dynamical systems on the rapidity of their approximations by the periodic dynamical systems which are simplest from some point of view. We shall see that many properties of dynamical systems are intimately related to the character of their approximations.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Chapter 16. Special Representations and Approximations of Smooth Dynamical Systems on the Two-dimensional Torus

Abstract
Suppose the space M is the two-dimensional torus Tor2 = ℝ2/ℤ2 with cyclic coordinates (u, v) and Lebesgue measure du dv. Consider the system of differential equations
$$\frac{{du}}{{dt}} = A(u,v),\quad \frac{{dv}}{{dt}} = B(u,v)$$
(1)
on it, with right-hand sides of class C r , r ≥ 2. This system satisfies the existence and uniqueness conditions and we may therefore introduce the one-parameter group {T t } of translations along its solutions.
I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai

Backmatter

Weitere Informationen