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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 76 77 78 79 80 81 < 82 > 83 84 85 86 87 88 .. 99 >> Next Theorem 8.15. Suppose T:KP -┬╗ Kr is an affine transformation. Tx = a Ō¢Ā. l'.\). where a e Kp anti A is a surjcctiie endomorphism of Kp. If m is Naar measure then
/i(T) = hjT) = liŌĆ×(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
ą½ąó) = hJT) = hJA) = ąØąø)
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. Ō¢Ī
┬¦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 NŌĆ×{T) will denote the cardinality of the set FŌĆ×(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 JVŌĆ×(T).
Proof. Let <5 be an exp insive constant for T. If T"x = v, TŌĆØy = čā and x čä čā then if d(TJ(x), TJ( r)) <6,0<j<n ŌĆö 1, then d{TJ(x), Tj(y)) <, 64j e Z and hence x = y. Therefore the set FŌĆ×(T) = Jx| T"x = x} is (n. S) separated and so NŌĆ×(T) < sŌĆ×(X,6) < oo. Hence
lim sup - log Nn(T) ^ lim sup - log.sŌĆ×(A',┬Ż) ^ h(T). Ō¢Ī
n-+nt ąś nŌĆö ą░čü 11
We are interested in the distribution of the periodic points so we čéą░ą║ąĄ 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 čĆąĄ M(X, V) describes the distribution of the periodic points of T if
N \T) I in M(X).
1 I xeFŌĆ×(T)
Theorem 8.17. Suppose ąó:ąźąø ŌĆö> .V., is a two sided topological Markov chain where A is an irreducible matrix Then h(T) ŌĆö IimŌĆ×^ , (l/┬╗i)log \ŌĆ×(T) and the wiiijiie measure nith maximal entropy describes the distribution of the periodic points of T.
Proof. If {čģ,}^* is a point of XA then it belongs toFŌĆ×(T) iff Xj- = vj4ŌĆ×Vj e Z. Therefore
NŌĆ×(T) = ┬Ż*
*o.....in- i^O
ą║
= trace of AŌĆØ = ┬Ż /"
i= 1
where ..., /.k are the eigenvalues of A. Therefore
ą╗-/ a" \ a"
where '/. is the largest positive eigenvaiuc of A which is simple because A is irreducible. Hence
lim 1 log iVŌĆ×(T) = log a = h(T). n ŌĆö / 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 JfdpŌĆ×-*\f dp for functions of the form / = yc where ąĪ = {{*;} 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 ŌĆö r
^f dpn = Y. airir + i ŌĆÖ ' ŌĆÖ ai.~ i ' ain*r-
1. ŌĆó ŌĆó ŌĆó t 1 = 0
_ airir + i ŌĆś ' ' ai.- i_)'.┬╗V Qirif┬½i*ŌĆś ai,-ii.UŌĆśrVi.
nŌĆ×(t) čÅ-";." )rr
by the above and Theorem 0.17. Therefore J/dpŌĆ× -┬╗Jf dp. Ō¢Ī
58.6 Definition of Measure-Thcore.ic Entropy Using the Mctrics <7ŌĆ×
205
A special case of this theorem says that the (1 ą║.....1 ą║) 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 ąÜ" which is expansive (i čü. [ąø] has no eigenvalues of absolute value 1). Then h{ 1) = IimŌĆ×^ T (1 n) log NŌĆ×(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  for the proof of the second part and to Bowen  and ┬¦22 of Denker, Grillenbcrger and Sigmund  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 ąÆ :KP -* Kp is an endomorphism of Kp onto Kp then the kernel of ąÆ contains |det [B]| points. (This is because the image B(I) of the unit square I of Rp under the linear map ąÆ 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" ŌĆö I : Kp -* Kp (using additive notation) is an endomorphism of A p onto Kp with corresponding matrix [An ŌĆö /] = [-4]ą╗ ŌĆö /.
Therefore det[/4" ŌĆö /] = , (A? ŌĆö 1) where Aj.....Xp are the eigenvalues
of [.4]. Then NŌĆ×(4) is the cardinality of the kernel of AŌĆØ ŌĆö I so NŌĆ×{A) = ą¤ąō=1 |/-ąō- 1| and
-logNŌĆ×(,4) = ┬Ż -log|k] - 1|.
If > 1 then
icg|;.? -1| = - [iog|;,|n 4- iog|i - ;.fn|] - iog|;.;|.
If p.,| < 1 then (l/ij)log|/," ŌĆö l| -┬╗0. Hence
lim ^ log NJtA)= ┬Ż log|Ai| = ąøąś). Ō¢Ī Previous << 1 .. 76 77 78 79 80 81 < 82 > 83 84 85 86 87 88 .. 99 >> Next 