# An introduction to ergodic theory - Walters P.

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(c) If xe В there are linear subspaces, {0} = V(0'(x) с= F(1)(x)

V«*»(x) = Rk, of Rk

(d) If xe В and 1 <, i ^ s(x) then limn_a. (l//i)log||Tx(T")r|| = л(,)(х) for all veV{i)(x)\V«-"(x).

(e) A(,,(x) is defined and measurable on {x e B|s(x) ;>/} and A(1,(Tx) = A<0(x).

(f) zx(T)V(‘\x) с: VM(Tx) if i < s(x).

The objects B, s, Vw do not depend on the choice of Riemannian metric.

If T is a diffeomorphism then A(I)(x) is never — oo and we also have xX(T) V{i){x) = V{,4Tx).

§10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem

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Remarks

(1) This theorem says there is a “large” subset of Л/ ("large" in the sense that m(B) = 1 Vhi e Л/(Л/, T)) such that the beha\ iour of ||гл(Т")г|| is known for all x e В and all v 6 тж\/. Onj can producc examples to show that one cannot obtain this result for all к e Л/. The functions /.<" arc called the Liapunov exponents of the dilTeomorphism T. As one might expect they are connected wuh the entropies of T (see Ruelle [3]).

(2) This result can be generalised to the case of a vector bundle map covering a consnuous map of a compact metric space (see Ruclle [4], Appendix D).

(3) There is also a version of Theorem 10.4 for difTeomorphisms that says the set of points С where the conclusion of Theorem 10.3 hold is large in the sense that m(C) = 1 Vw e MyM. T).

(4) If ;.("(x) < 0 then for every 0 ф v e l/(,)(.v) and every г > 0 we have ||тх(Ти)1[| < fol- aij large n. Therefore if /.*(x)i < 0 then all elements of K(1,(\) converge exponentially to zero with rate at most t ' under the iterates of the ungent map to T. Similarly if /,J,(\) > 0 then all elements of txXt V(J~ *'(x) have lengths which become infinite exponentially with rate at least under the iterates ofr(T). If/.(p,(x) = 0 then we know the elements of К(р'(х)' Klp-U(x) do not converge exponentially to 0 or oo under the iterates of z(T).

Suppose T is a diffcomorphism and consider the decomposition тXM = W(1)(x) © • • • © И'(3(х),(ч) over the “large” set С (see Remark (3)). Let </(x) be such that ;.'4<*»(х)<0<л,,(*,+1,(х) and p(x) such that A(pU,_ !)(x) < 0 < ;.<pU')(x). Either p(x) = ij(x) + 1 or p(x) = <j(a) + 2. Put Uc(x) = {0} in the first case and Uc(x) = H/,,(Jt)+ n(x) in the second case, and

«(x) J(X)

U‘(x) = © w(,)(x), Uu(x) = © И/°'(х).

i = I j = pM

(Note that Us(x) = K(,(JI),(x) in the previous notation.) Then тXM = Us(x)@ Uc(x) © Uu(x) and for each v Ф 0,ve U*(x), ||tx(T")i;|| goes to 0 with exponential rate at most A,,uW(x); for each v ф 0, v e Uu(x), ||rx(Tn)r|| goes to oo with exponential rate at least /.(p(Jt,)(x); and for each v Ф 0, v e Uc(x), [|тх(7’п)г|| does not converge to 0 or oo with any exponential rate. It is the contraction caused by the “stable” subspaces l/s(x) and the expansion determined by the “unstable” subspaces Uu(x) that make the interesting behaviour of T:M -» M.

Let us now consider the interpretation of the above results for two examples.

Consider first the north-south difieomorphism T: К -» К of the unit circle. We know M(K, T) consists of all convex combinations of 6N, iK (where N is the north pole of the circle and S is the south pole) so for a set В e л(К) to havem(B) = 1 V m e M{K, T) means that N and S belong to В The set {N,S}, although only containing two members, is large from the point of view of

236

10 Applications and Other Topics

the dynamics of T because we know that all other points move away from Л' towards S under iteration of T. The tangent bundle xk can be considered as the product К x R and we can write x(T):K x R-> К x R by r(T)(z, v) — (T(:), T'(z) ■ v) where T':K R is a function. Therefore x(T")(x,v) — (T"(r), T\Tn~1z) ■ • ■ T'(z)i-) so if ||-|| denotes the trivial Riemannian metric on К (||U,r)|| = |i|) then

- log||r(T'-)(2,r)|| = - £ log(|T'(T‘-)| ■ |ii|) - log(|T'(s)|) if г Ф N, n n i = 0

because T"(z)-*S and hence log]T’( Г";)) -»log|T'(5)|. So in lact we could take В = К in Theorem 10.4 and /("(Ar) = lo«|7“'(Ar)| > 0, P"(/V) = and if z ф Лг, /.<u(~) = log|T'(S)| < 0. V*w(«) = x.K. This shows the function /.u> need not be continuous.

Now let A: Kp Kp be an automorphism of the p-dimersional torus. The tangent bundle xKp can be represented as the product Kp x Rp and then the tangent map xA: Kp x Rp -» Kp x Rp is given by t(.4)(.\\ r) = (.4(.\). Av) where A:RP -* Rp is the linear transformation that covers A. Take the Riemannian metric where ||(.v,r)|| is the Euclidcan length of ve Rp. Then ||т(/Г)(л,r)|| = ||(y4"(.v)ki4"(«J.)|| = ||<4n(r)|| Vh ^ 1. Therefore if л(М < /.a> < - • • < /<5' are the numbers such that fA<",..., f;<“ are the distinct absolute values of eigenvalues of A and l/(1> cz • ■ • cz K(5) = Rp is the corresponding filtration of Rp then, for all x. s(x) = s, /."’(x) = л"’ and F"'(.v) = {jc] x Vм.

We know that for each difTeomorphism T:M -* M we have a nice theory of expansion and contraction for the linearised situation т(Т)тЛ/ —> xM. To tackle problems about the action of T on M it is desirable to have a nonlinear version of this theory on M. Let us consider contraction. In the linearised situation this is determined by the subspaces K{,(JI))(.v) of xxM for x e В where q(x) is the largest natural number with /.(4<x,,(.v) < 0 (put q(x) = 0 if no such natural number exists). One would like to “integrate” the family .ve B) of subspaces by finding a family of smooth submanifolds of M which are disjoint and such that for each xe В the space Hwwrft(.\) is the tancent space to the submanifold containing x. This can be done and we refer the reader to Ruelle [4] or Pesin ([ 1 ], [2]) for the statement of such a theorem. An infinite dimensional version of this theory appears in Ruelle [5]. See Ruelle

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