# An introduction to ergodic theory - Walters P.

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Theorem 3.6 (Representation Theorem). An ergodk measure-preserving transformation T with discrete spectrum on a probability space (X,J$,m) is . conjugate to an ergodic rotation on some compact abelian group. The group

■ will be metrisable iff (Ahas a countable basis.

Proof. Let A be the group of all eigenvalues of T and give A the discrete topology. So A is an algebraic subgroup of К but has me discrete topology. I If L2(jii) is separable then A is countable). Let G = A, the character group of A. Then G is compact and abelian. By the duality theorem (3 of §0.7) G = A is naturally isomorphic to A. For ле A we shall let л denote the corresponding element of G i.e. /.(g) = g(/.) Mg e G = A. The map a: A —> К given by a(k) = A is a homomorphism of the discrete group A into К and so belongs to A = G. Therefore /.(a) = а(л) = /. V/. e A.

Define S.G -* G by S(g) = ay. Wc chum that S is ergodic. Let p denote Haar measure on G and suppose / ° S = /, / e L2(p). Then / has a Fourier series / = £,■ bj/.r A,eA. From / S=f we have = Y-i

/’/i-jffl) so bjkj(a) = br Since ЯДа) = /., this gives bf/.j = bj. If bj ф 0 then кi = 1 so that /.j = 1. Therefore the only non-vanishing term in the Fourier series is the constant term and so / is constant a.e. Wc now know S is ergodic and, by Theorem 3.5, it has discrete spectrum.

Again by Theorem 3.5 the group of eigenvalues'1 of S is {y(a):y e G} = !«(л):л e /1} = {а:л e A} = A. Therefore S and T have the same eigenvalues and both have discrete spectrum. By the Discrete Spectrum Theorem they are conjugate.

The group G is metrisable liT A is countable and this is equivalent to L2(m) being separable. □

Theorem 3.7 (Existence Theorem). Every subgroup A of К is the group of eigeniatues of an ergodic measure-preserving transformation with discrete .sipeel rum.

Proof. The desired transformation is the rotation S constructed in the proof of Theorem 3.6. □

The conjugacy problem for ergodic measure-preserving transformations with discrete spectrum is completely solved. We have some very simple invariants, namely the eigenvalues, which determine when two such transformations are conjugate. Each conjugacy class of ergodic measure-preserving transformations with discrete spectrum is characterized by a subgroup of к and each subgroup of К corresponds to a conjugacy class. So for this class of transformations we get a very satisfying solution to the .conjugacy problem. Also there arc some simple examples, namely group

У Measure-Preserving Transformation}, with Discrete Spccirum

reunions, such that each ergodic measure-preserving transformation with discrete spectrum is conjugate to one of these examples

An extension of taese results to a wider class of transformations has been carried out by Abramov [1]. Hoare and Parry [1] and Helm and Parry [1], [2]. This class is the collection of all transformations with quasi-discretc spectrum. The invariant is not just one group but a sequence of groups connected by homomorphisms. The canonical examples in this class are certain affine transformations of compact abelian groups.

CHAPTER 4

Entropy

We jre searching for conjugacy and/or isomorphism invariants. In 1958 Kolmogorov introduced the concept of entropy into ergodic theory, and this has been the most successful invariant so far. For example, in 1943 it was known that the two-sided (-j,i)-shifT and the two-sided (3,7, -j)-shift both have countable Lebesgue spectrum and hence are spectrally isomorphic, but it was not known whether they are conjugate. This was resolved in 1968 when Kolmogorov showed that they had entropies log 2 and log 3, respectively, and hence are not conjugate. The notion of entropy now used is slightly difTerent from that used by Kolmogorov—the improvement was made by Sinai in 1959.

The definition of the entropy of a measure-preserving transformation T of (Ais in three stages: the entrop^6f a finite sub-с-algebra of Jtf, the entropy of the transformation Tj;elative to a finite sub-<7-algebra, ano, finally, the entropy of T. Each stage of the definition is quite simple to state. Before giving the definition we shall study finite sub-<7-algebras of /А and give some motivation for the definition.

The definitions involve logarithms and we shall use natural logarithms. This is because it will be more natural in Chapter 9, to tie in with some ideas from statistical mechanics. Some authors use logarithms of base 2.

§4.1 Partitions and Subalgebras

Throughout this chaptcr (Awill denote a probability space.

Definition 4.1. A partition of {X,.JAsm) is a disjoint collection of elements of /M whose union is X.

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7(1

4 Entropy

We shall be ini с rested in unite partitions. The) will be denoted by Greek

letters, e.g.. с = {.-I,.....-lfc|.

If С is a finite partition o' (X,?}.m) then the collection of all elements of /,’ \\ hich are unions of elements of с is a finite siib-cr-alarbra of У1. Wc denote it by -/к). Conversely, ifis a finitc^tib-rr-algebra of .*?, say 7Г = [f6;:i =

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