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An introduction to ergodic theory - Walters P.

An introduction to ergodic theory

Author: Walters P.
Publishers: London
Year of publication: 1982
Number of pages: 251
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Download: anintroduction1982.djvu

CHAPTER О
Preliminaries
§0.1 Introduction
In its broadest interpretation ergodic theory is the study of the qualitative propci lies of actions of groups on spaces. The space has some structure (e.g. the space is a measure space, or a topological space, or a smooth manifold) and each element of the group acts as a transformation on the space and preserves the given structure (e.g each element acts as a measure-preserving transformation, or a continuous transformation,or a smooth transformation).
To see how this type of study arises consider a system of к particles moving in R3 under known forces. Suppose that the state of the system at a given time is determined by knowing the positions and the momenta of each of the к particles. Thus at a given time the system is determined by a point in R6k. As time continues the system alters according to the differential equations governing the motion, e.g., Hamilton’s equations
dq-t _ cH dpi _ oH dt dp^ dt dqt'
•“W
If we are given an initial condition and if the equations can be uniquely solved then the corresponding solution gives us the entire history of the system, which is determined by a curve in R6k.
If x is a point in the state space representing the system at a time t0, let T,(x) denote the point of the state space representing the system at time t + t0. From this we ьее that T, is a transformation of the state space and, moreover, T0 = id and T,TJ =_T, о Ts. Thus t-*T, is an action of the group R on the state space. Because the Hamiltonian h is constant along solution curves, each energy surface H~l(e) is invariant for the transformation T,
1
2
0 I'rdiiiiiiKines
so that wc gel an action of R on each energy surface. One is interested in the asymptotic properties of the action i.e. in T, for large I. The transformations T, И~'(с) are continuous and arc smooth if H~ ‘(t') ij smooth. Measure theory enters this picture via a theorem of Liouville which tells us that if the forces arc of a certain type one can choose coordinates in the state space so that the usual 6A-dimensional measure in these coordinates is preserved by each transformation T,.
The word “crgodic” was introduced by Boltzmann to describe a hypothesis about the action of -|r,jf £ R) on an energy surface when the Hamiltonian И is of the type that arises in statistical mechanics. Boltzmann had hoped that each orbit {T,(.v)| t e R\ would equal the whole energy surface H~‘w imd he called this statement the ergodic hypothesis. The Word “ergodic” is an amalgamation of the Greek words ergon (work) and odos (path). Boltzmann made the hypothesis in order to deduce the equality of time means and phase means which is a fundamental algorithm in statistical mechanics. The ergodic hypothesis, as stated above, is false. The property the flow needs to satisfy in order to equate time means and phase means of real-valued functions is what is now calico crgodicity.
It is common to use the name ergodic theory to describe only the qualitative study of actions of groups on measure spaces The actions on topological spaces and smooth manifolds are often called topological dynamics and differentiable dynamics. This measure theoretic stud\ began in the early 1930’s and the crgodic theorems of BirkhofTand von Neumann were proved then. The next major advance was the introduction of entropy by Kolmogorov in 195S. The proof, by Ornstein in 1969, that entropy was complete for Bernoulli shifts revitalised the work on the isomorphism problem. During recent years crgodic theory had been used to give important results in other branches of mathematics.
We shall study actions of the group Z of integers on a space X i.e. we study a transformation T\X~* X and its iterates T", n e Z. This is simpler than studying (he actions of R. Of course, if {T, | r e R) is an action of R on X, then by choosing (0 Ф 0 and observing the system at the times ..., — (0, 0, t0, 2t0, 3f0.....wc arc considering (T,Jn, n e Z.
In the follow mg sections wc summarise some of the background ideas and notation we shall be using.
We shall use 7 to denote the set of integers, Z^ to denote the non-negative integers, R to denote the real numbers, R* to denote the non-negative reals, and С to denote the complex numbers. The empty set will be denoted by 0.
If A,В are subsets of a set A", then B\A denotes the difference set {x e X |,v e В, x ф A}, and А Д В denotes the symmetric difference (A\B) vj (£' /1). We use 2* to denote the collection of all subsets of X.
We use “ilT" to denote "if and only if.” We number lemmas and theorems in a single sequence (Theorem 5.6 is the sixth theorem in Chapter 5) but giv^ , corollary the s.une number as the corresponding theorem (Corollary 5.6.2' is the second corollary of Theorem 5.6).
vj0.2 Measure Spaces
з
§0.2 Measure Spaces
We shall generally refer to Kingman and Taylor [1] and Parthasarathy [2]. Let X be a sci. A a-algebra of subsets of X is a collection SO of subsets of X satisfying the follqwing фгее conditions :(i)A g £4?; (ii) if В ей? then X\B edl\
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