# An introduction to ergodic theory - Walters P.

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Another invariant of conjugacy for non-invertiblc transformations is the decreasing sequence of <7-aIgebras {However the three invariants of eniropy . Jacobian and the sequence \ T~H A | ,'„0 are not complete for ihe relation of conjugacy on ihe class of exact endomorphisms because there ;:ге two exact endomorphisms Л’, T wilh S~"4l — T~nJ£ n > 0, S2 = T2 i=* InS) = h\T)), S and T having equal Jacobians but with S and T not conjugate.

Also a one-sided Markov chain which is exact need noi be conjugate to a one-sided Bernoulli shift (Parry and Wallers [1]).

§4.13 Comments

Entropy was introduced as a conjugacy invariant for measure-preserving transformations. It was soon realized that entropy iheory was more than just an assignment of a number to each transformation. Kolmogorov automorphisms and transformations with zero entropy have received ihe mosi treatment. They are “opposites” from the point of view of entropy. Kolmogorov automorphisms are important for applications as it seems that ihe most interesting smooth systems are Kolmogorov and even Bernoulli.

By Theorem 4.36 we know that the spectral theory of invertible measure-prcsjrving transformations reuuccs to that for the zero eniropy case. Th^ following is still an open problem: If /i(T) = 0 whai kind of spectrum can U г have?

For transformations with zero entropy the isomorphism problem is only solved for ergodic transformations with discrete spectrum, totally ergodic transformations with quasi-discrete spectrum and some other special cas^s. Sequence eniropy may play a role in the isomorphism problem for zero entropy transformations

In tne weak topology on the set of all invertible measure-preserving transformations on a given space the set of transformations with

^4 1 ,i Comments

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zero entropy is a dense Ctj (countable intersection of open sets) jRohlin [5]). Since the set of weak mixing transformations is also a dense G(5 and the set of strong mixing transformations is a set of first category it follow^ that "most” transformations are weak mixing have zero entropy, but are not strong mixing. However the ones of interest for applications are often not in this class.

The main problem to consider for Kolmogorov automorphisms seems to be io find more examples of Kolmogorov automorphisms that are not Bernoulli automorphisms. One should first check whether the usual ways of constructing new- transformations from old ones transform a Bernoulli automorphism to a Bernoulli automorphism (e.g., is a weak mixing group extension of a Bernoulli automorphism a Bernoulli automorphism?). Several results in this direction are known. If one of these constructions leads to a Kolmogorov automorphism which is not a Bernoulli automorphism then this may lead to a new invariant that may be complete for Kolmogorov automorphisms.

CHAPTER 5

Topological Dynamics

In measure theoretic ergodic theory one studies the asymptotic properties of measure-preserving transformations. In topological dynamics one studies the asymptotic properties of continuous maps. We shall study continuous transformations of compact metric spaces. The compactness assumption is a “finiteness” assumption which is similar to the assumption of a finite measure in the measure-theoretic work. The assumption of metrisability is not needed for some of the results but it often shortens proofs and most applications are for metric spaces. The notations we shall use are given in §0.10.

If X is compact metric and T:X -* X is continuous one has an induced map UT:C(X) C(X) given by Urf = / ° T. The map Ur is clearly linear and multiplicative (i.e. Ur(f ■ g) = (UTJ)(Ury')). If T maps X onto X then UT is an iaometry and if Г is a homeomorphism then Ur is an isometric automorphism in the sense of Banach algebras (i.e. a multiplicative linear isometry of C(X) onto C(X)).

In the first section we give a list of examples and in subsequent sections we discuss dynamical properties. We shall connect these properties with measures in Chapter 6, when we study the family of invanant probability measures for a given continuous transformation.

§5.1 Examples

(1) The identity, I, on any X.

(2) A rotation Tx = ax on a compact metric group (recall from §0.6 that such a group has a rotation invariant metric).

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§5.1 Examples

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(3) A surjective endomorphism of a compact metric group; in particular, of a torus.

(4) An affine transformation Tx = а - Л(х) where A is a surjective endomorphism of a compact group G and oeC. This example includes examples 2 and 3.

(5) Let У = {0, 1,..., к — 1} with the discrete topology. Let X = П-® Y with the product topology. A neighbourhood basis of a point {x„} consists of the sets UN — {{^n} | — xrLf°r И ^ N}> N ^ 1. A metric on X is given by

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The two-sided shift T, defined by T{x„} = {y„} with yn — xn+1, is a ho-meomorphism of X. We sometimes write this T{..., x_ tx0xu ...) = (... ,x_iX0XjX2, ...) where the symbol * occurs over the 0-th coordinate of each point. Note that here we have a special case of (iii) since X is a compact group under the operation

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