# An introduction to ergodic theory - Walters P.

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S6.6 Examples

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and p = (p„, ..., pk _,) is a probability vector with ^ <! PiPij = Pj the corresponding Markov measure belongs to M(X.T\. The product measures are special examples of Markov measures (p;J = pj. Atomic measures in M(X. T) .jre proviaed by Theorems 5.12 and 6 16.

(6) When T\t\ ■* К is the north-south nutp know 1ЦТ) — {/V,но that, by Theorem 6.15 A/(K, T) = lpSN + (1 — p)os\p 6 [& 1 J}-

(7) Suppose T.X A -» A'., is a two-sided topological Markov chain, where A — («ij)?j=o is a к x к matrix with ai} e [0,1J. The set М(ХЛ, T) depends very much on A; when au = 1 Vi,j then we have Example 5, and when A = I then T is the identity map on a space with к points. Howe\cr if P = (pt) is a к x к stochastic matrix with 0 < pu < atj all i, j and p = (p0,..., ,) is a probability vector with У-_'d р,р^ = pj the Markov measure determined b) p and P is a member of Л/(Л'T) because il gives /его measure to Л' XA. When A is irreducible (i.e. Vi, j there is some n = n(i,j) such that A" has (/, j)-th element non-zero) we can obtain such a member as follows. By the Perron-Frobenius theory of non-negative matrices (see ij0.9) there is /_ > 0 which is an eigenvalue of A and no other eigenvalue of A has larger absolute value. If A is irreducible ihen '/. is a simple eigenvalue and the corresponding right and left eigenvectors have strictly positive entries. Suppose ]Г; = о u,u,j = /.Uj and Yj=o aijvj — where u, > 0. vt > 0 all /. Normalise (ut,. .. ,uj and (i-„ .. ,i„) so that i/,r, = 1. Put pt = u,i\ and p^ = a,-///,v,. Then P = (Pij) is a stochastic matrix, 0 < p{j < ay, and Yj = o Wfij = Py Tiierefore the Markov measure determined by (p0.... ,pk-t) and P is a member of M(XA, T). We shall see in Chapter 8 that this is a very important member of

M(XA,n

We now know that every continuous transformation T : X -> X of a compact metric space has an invariant probability defined on the Borel subsets of X. For some transformations, such as toral automorphisms and shift homeomorphisms, the space M(X, T) contains many elements. The question arises as to which are the important elements of M(X. T) to study. It would be good if we could characterise certain members of M(X,T) by “physical principles” (such as variational principles) and that these measures had strong ergodic properties (such as making Ta Bernoulli automorphism). We shall see in Chapters 8 and 9 that such a variational principle ex'sts (and is analogous to a well known variational principle in statistical mechanics). It turns out that for some transformations the measures picked out by the variational principle do have strong crgodic properties. The variational principle uses the idea of topological entropy which we discuss in the next chapter.

CHAPTER 7

Topolcgical Entropy

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Adler, Konheim, and McAndrew [1] introduced topological entropy as :m invariant of topological conjugacy and also as an analogue of measure theoret.c entropy. To each continuous transformation T:X -» X ofacompact topological space a non-negative real number or oc, denoted by h(T), is assigned. Later Dinaburg and Bowen gave a new, but equivalent, definition and this definition led to proofs of the results connecting topological and measure-theoretic entropies. Bowen defined the entropy of a uniformly continuous map of a (not necessarily compact) metric space and this leads — to a geometuc proof of the formula for the topological entropy of an automorphism of an /i-torus. In the first section of this chapter v\e give the original definition of topological entropy and in §7.2 we give the other definition. In the last section we calculate the topological entropy of our examples.

We shall use natural logarithms because this will be more appropiiate when we discuss topological pressure in Chapter 9.

§7. PDefinition Using Open Covers

Let X be a compact topological space. We shall be interested in open covers of X which we denote by a, /?,

Definition 7.1. If a, p are open covers of X their join a v /? is the open cover by all sets of the form A n В where A e а, В e Similarly wc can dcline the join \Jl=, af of any finite collection of open covers of X.

Definition 7.2. An open cover /? is a refinement of an open cover a, written a < /?, if every member of /? is a subset of a member of a.

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§7.1 Definition Usin^ Open Covers

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Hence a < av/Lfor any open covers a, /1 Also if P is a subcover of a then a < p.

Definition n.3. If a js an open cover of X and T:X -» X is continuous then T~1 a is the open cover consisting of all sets T~1 A where Леа.

T ‘(7vfl = T ‘(a)vT and а < ft implies T 'з < T xp.

We shall denote avT"1av---vT'l""1]Kby Vi = o T~‘a.

Definition 7.4. If a is an open cover of X let N[ci) denote the number of sets in a finite subcover of a with smallest cardinality. W'e define the entropy of a by H(a) = logN(a).

Remarks

(1) H(s) > 0

(2) W(a) = 0 iff ЛЧа) = 1 iff X e a.

(3) If а < P then H(a) < H( P).

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