# An introduction to ergodic theory - Walters P.

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Theorem 8.15. Suppose T:KP -» Kr is an affine transformation. Tx = a ■. l'.\). where a e Kp anti A is a surjcctiie endomorphism of Kp. If m is Naar measure then

/i(T) = hjT) = li„(A)= h(A) = Y. log|/.j|,

!i:|iX> II

where /,.....kp are the eiyemallies of the Matrix [\-l] which represents A.

Proof. We know by Theorem S I 1 that

ЫТ) = hJT) = hJA) = НЛ)

and by Corollary 8.12.1 that /?(. t j = /i( -0, where A denotes the covering linear map of A. Since -1 is represented by the matrix [ 1] in the natural basis the formula for h( 5) follows from Theorem 8.14. □

§8.5 The Distribution of Periodic Points

If a continuous map T X -* X has a unique measure w ith miximal entropy one expects this measure to have strong properties and tie in with the other dynamical behaviour of T. We discuss how, for some maps T, thk measure is connected with periodic points.

If T: X -* X is a continuous transformation of a compact metrisable space then N„{T) will denote the cardinality of the set F„(T) = |.v e X|T"(\) = x). We have

Theorem 8.16. If T: X -* X is an expansive homeomorphism of a compact metric space then Nn(T) < x Vn 1 and h(T) 2: limsupnJ (1 />»)log JV„(T).

Proof. Let <5 be an exp insive constant for T. If T"x = v, T”y = у and x ф у then if d(TJ(x), TJ( r)) <6,0<j<n — 1, then d{TJ(x), Tj(y)) <, 64j e Z and hence x = y. Therefore the set F„(T) = Jx| T"x = x} is (n. S) separated and so N„(T) < s„(X,6) < oo. Hence

lim sup - log Nn(T) ^ lim sup - log.s„(A',£) ^ h(T). □

n-+nt И n— ас 11

We are interested in the distribution of the periodic points so we таке the following definition.

204

8 Relationship Between Topological Entropy and Measure-Theoretic Untropv

Definition 8.4. Let T:X -* X be a continuous map of a compact metiisable space with Nn( T) < x V»i ;> 1. A measure ре M(X, V) describes the distribution of the periodic points of T if

N \T) I in M(X).

1 I xeF„(T)

Theorem 8.17. Suppose Т:ХЛ —> .V., is a two sided topological Markov chain where A is an irreducible matrix Then h(T) — Iim„^ , (l/»i)log \„(T) and the wiiijiie measure nith maximal entropy describes the distribution of the periodic points of T.

Proof. If {х,}^* is a point of XA then it belongs toF„(T) iff Xj- = vj4„Vj e Z. Therefore

N„(T) = £*

*o.....in- i^O

к

= trace of A” = £ /"

i= 1

where ..., /.k are the eigenvalues of A. Therefore

л-/ a" \ a"

where '/. is the largest positive eigenvaiuc of A which is simple because A is irreducible. Hence

lim 1 log iV„(T) = log a = h(T). n — / n

Let p denote the unique measure with maximal entropy (see Theorem 8.10; we use the same notation as there). To show

^" = nTh ^ h

'vnM I xet'x(T)

it suffices to show Jfdp„-*\f dp for functions of the form / = yc where С = {{*;} I * |xr = ir,..., x, = is} for some r, s (r < s) and some , is. This is because finite linear combinations of such functions arc dense in C(XA) by the Stone-Weierstrass theorem. Let / = yc be as above. If n > s — r

^f dpn = Y. airir + i ’ ' ’ ai.~ i ' ain*r-

1. • • • t 1 = 0

_ airir + i ‘ ' ' ai.- i_)'.»V Qirif«i*‘ ai,-ii.U‘rVi.

n„(t) я-";." )rr

by the above and Theorem 0.17. Therefore J/dp„ -»Jf dp. □

58.6 Definition of Measure-Thcore.ic Entropy Using the Mctrics <7„

205

A special case of this theorem says that the (1 к.....1 к) product measure

describes the distribution of the periodic points of the two-sided shift.

The corresponding result holds for automorphisms of tori:

Theorem 8.18. Suppose A:KP-*KP is tin autnmorphhm of К" which is expansive (i с. [Л] has no eigenvalues of absolute value 1). Then h{ 1) = Iim„^ T (1 n) log N„(A) and the Haar measure m describes the distribution oj the periodic points of A. (The same conclusions hold when A is merely ergodic but the proof is more delicalc.)

We refer to Bowen [4] for the proof of the second part and to Bowen [5] and §22 of Denker, Grillenbcrger and Sigmund [1] for a discussion of generalisations to a wider class of homeomoiphisms. We now prove that h{A) = limn^ar(l >i)logNn(A).

Proof. In the proof wc shall use the fact that if В :KP -* Kp is an endomorphism of Kp onto Kp then the kernel of В contains |det [B]| points. (This is because the image B(I) of the unit square I of Rp under the linear map В has Lebesgue measure |det [B] ind so when B(I) is reduced mod Zp each point of 1 is covered by |det[B]| points of B(/)).

Since A is expansive. A" — I : Kp -* Kp (using additive notation) is an endomorphism of A p onto Kp with corresponding matrix [An — /] = [-4]л — /.

Therefore det[/4" — /] = , (A? — 1) where Aj.....Xp are the eigenvalues

of [.4]. Then N„(4) is the cardinality of the kernel of A” — I so N„{A) = ПГ=1 |/-Г- 1| and

-logN„(,4) = £ -log|k] - 1|.

If > 1 then

icg|;.? -1| = - [iog|;,|n 4- iog|i - ;.fn|] - iog|;.;|.

If p.,| < 1 then (l/ij)log|/," — l| -»0. Hence

lim ^ log NJtA)= £ log|Ai| = ЛИ). □

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