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## Über dieses Buch

This version differs from the Portuguese edition only in a few additions and many minor corrections. Naturally, this edition raised the question of whether to use the opportunity to introduce major additions. In a book like this, ending in the heart of a rich research field, there are always further topics that should arguably be included. Subjects like geodesic flows or the role of Hausdorff dimension in con­ temporary ergodic theory are two of the most tempting gaps to fill. However, I let it stand with practically the same boundaries as the original version, still believing these adequately fulfill its goal of presenting the basic knowledge required to approach the research area of Differentiable Ergodic Theory. I wish to thank Dr. Levy for the excellent translation and several of the correc­ tions mentioned above. Rio de Janeiro, January 1987 Ricardo Mane Introduction This book is an introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems, which is sometimes called differentiable ergodic theory. Chapter 0, a quick review of measure theory, is included as a reference. Proofs are omitted, except for some results on derivatives with respect to sequences of partitions, which are not generally found in standard texts on measure and integration theory and tend to be lost within a much wider framework in more advanced texts.

## Inhaltsverzeichnis

### Chapter 0. Measure Theory

Abstract
The purpose of this chapter is to present the basic definitions and theorems of measure theory. Proofs are only included when they cannot be found in standard references or when the formulation of the statement involved differs significantly from the usual one. The basic references for this chapter are Rudin [R6] and Munroe [M13].
Ricardo Mañé

### Chapter I. Measure-Preserving Maps

Abstract
If (X, A, μ) is a measure space, we say that a measurable map T: XX is measure-preserving, and that μ is invariant under T, if for every AA we have μ(A)= μ(T−1(A)). The dynamic behavior of measure-preserving maps is the theme of ergodic theory.
Ricardo Mañé

### Chapter II. Ergodicity

Abstract
Around the turn of the century the work of Boltzmann and Gibbs on statistical mechanics raised a mathematical problem which, in our context, can be stated as follows: given a measure-preserving map of a space (X, A, μ) and an integrable function f: XR, find conditions under which the limit
$$\mathop {\lim }\limits_{n \to + \infty } \frac{{f\left( x \right) + f\left( {T\left( x \right)} \right) + \cdots + f\left( {{T^{n - 1}}\left( x \right)} \right)}}{n}$$
(1)
exists and is constant almost everywhere. Similar questions had already shown up in other areas of mathematics, for example, in the problem of the average movement of the perihelion in celestial mechanics (see Arnold [A6]).
Ricardo Mañé

### Chapter III. Expanding Maps and Anosov Diffeomorphisms

Abstract
In Section I.1 we highlighted the fact that some maps have an invariant measure naturally associated with them, but that ergodic theory also comprises, among its aims, the dynamical study of maps which are not born with an associated invariant measure. The first step in this direction was showing that every continuous map of a compact metric space has an invariant measure (I.8). The second was the ergodic decomposition theorem and the concept of regular points (II.6). Not much more can be said about continuous maps in general; in order to develop our theory further, we will restrict our attention to differentiable maps of closed manifolds. Before doing that, however, we will introduce the concept of expanding maps of metric spaces—roughly, maps which locally increase distances. For such maps, it is possible to find, among all invariant measures, one whose properties make it especially interesting. We start with two particular cases of the definition, where the results obtained are particularly relevant.
Ricardo Mañé

### Chapter IV. Entropy

Abstract
In Section I.2 we introduced one of the fundamental problems of Ergodic Theory, namely, deciding when two automorphisms T1, T2 of probability spaces (X1 , A1, μ1) and (X2, A2, μ2)are equivalent. The approach developed there, involving the study of spectral properties of the associated isometric operators $${U_{{T_i}}}$$: L2 (X i , A i , μ i ) → L2 (X i , A i , μ i ) (i = 1,2), led to the concept of spectral equivalence. We proved that equivalent maps are spectrally equivalent, and mentioned that the converse is false.
Ricardo Mañé

### Backmatter

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