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# An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 87 88 89 90 91 92 < 93 > 94 95 96 97 98 .. 99 >> Next (c) If xe ąÆ there are linear subspaces, {0} = V(0'(x) čü= F(1)(x)
V┬½*┬╗(x) = Rk, of Rk
(d) If xe ąÆ and 1 <, i ^ s(x) then limn_a. (l//i)log||Tx(T")r|| = ą╗(,)(čģ) for all veV{i)(x)\V┬½-"(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(ŌĆś\x) čü: 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 ŌĆö oo and we also have xX(T) V{i){x) = V{,4Tx).
┬¦10.2 The Subadditive Ergodic Theorem and the Multiplicative Ergodic Theorem
235
Remarks
(1) This theorem says there is a ŌĆ£largeŌĆØ subset of ąø/ ("large" in the sense that m(B) = 1 Vhi e ąø/(ąø/, T)) such that the beha\ iour of ||ą│ą╗(ąó")ą│|| is known for all x e ąÆ and all v 6 čéąČ\/. Onj can producc examples to show that one cannot obtain this result for all ą║ e ąø/. 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 ).
(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 , Appendix D).
(3) There is also a version of Theorem 10.4 for difTeomorphisms that says the set of points ąĪ 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 čä v e l/(,)(.v) and every ą│ > 0 we have ||čéčģ(ąóąĖ)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 ąÜ(čĆ'(čģ)' 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 čéXM = W(1)(x) ┬® ŌĆó ŌĆó ŌĆó ┬® ąś'(3(čģ),(čć) over the ŌĆ£largeŌĆØ set ąĪ (see Remark (3)). Let </(x) be such that ;.'4<*┬╗(čģ)<0<ą╗,,(*,+1,(čģ) 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
┬½(x) J(X)
UŌĆś(x) = ┬® w(,)(x), Uu(x) = ┬® ąś/┬░'(čģ).
i = I j = pM
(Note that Us(x) = K(,(JI),(x) in the previous notation.) Then čéXM = Us(x)@ Uc(x) ┬® Uu(x) and for each v ąż 0,ve U*(x), ||tx(T")i;|| goes to 0 with exponential rate at most A,,uW(x); for each v čä 0, v e Uu(x), ||rx(Tn)r|| goes to oo with exponential rate at least /.(p(Jt,)(x); and for each v ąż 0, v e Uc(x), [|čéčģ(7ŌĆÖą┐)ą│|| does not converge to 0 or oo with any exponential rate. It is the contraction caused by the ŌĆ£stableŌĆØ subspaces l/s(x) and the expansion determined by the ŌĆ£unstableŌĆØ 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: ąÜ -┬╗ ąÜ 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 ąÆ e ą╗(ąÜ) to havem(B) = 1 V m e M{K, T) means that N and S belong to ąÆ 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 ąø' towards S under iteration of T. The tangent bundle xk can be considered as the product ąÜ x R and we can write x(T):K x R-> ąÜ x R by r(T)(z, v) ŌĆö (T(:), T'(z) Ō¢Ā v) where T':K R is a function. Therefore x(T")(x,v) ŌĆö (T"(r), T\Tn~1z) Ō¢Ā ŌĆó Ō¢Ā T'(z)i-) so if ||-|| denotes the trivial Riemannian metric on ąÜ (||U,r)|| = |i|) then
- log||r(T'-)(2,r)|| = - ┬Ż log(|T'(TŌĆś-)| Ō¢Ā |ii|) - log(|T'(s)|) if ą│ ąż N, n n i = 0
because T"(z)-*S and hence log]TŌĆÖ( ąō";)) -┬╗log|T'(5)|. So in lact we could take ąÆ = ąÜ in Theorem 10.4 and /("(Ar) = lo┬½|7ŌĆ£'(Ar)| > 0, P"(/V) = and if z čä ąøą│, /.<u(~) = log|T'(S)| < 0. V*w(┬½) = 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 ||čé(/ąō)(ą╗,r)|| = ||(y4"(.v)ki4"(┬½J.)|| = ||<4n(r)|| Vh ^ 1. Therefore if ą╗(ą£ < /.a> < - ŌĆó ŌĆó < /<5' are the numbers such that fA<",..., f;<ŌĆ£ are the distinct absolute values of eigenvalues of A and l/(1> cz ŌĆó Ō¢Ā ŌĆó cz K(5) = Rp is the corresponding filtration of Rp then, for all x. s(x) = s, /."ŌĆÖ(x) = ą╗"ŌĆÖ and F"'(.v) = {jc] x Vą╝.
We know that for each difTeomorphism T:M -* M we have a nice theory of expansion and contraction for the linearised situation čé(ąó)čéąø/ ŌĆö> 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 ąÆ 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 ŌĆ£integrateŌĆØ the family .ve B) of subspaces by finding a family of smooth submanifolds of M which are disjoint and such that for each xe ąÆ the space Hwwrft(.\) is the tancent space to the submanifold containing x. This can be done and we refer the reader to Ruelle  or Pesin ([ 1 ], ) for the statement of such a theorem. An infinite dimensional version of this theory appears in Ruelle . See Ruelle Previous << 1 .. 87 88 89 90 91 92 < 93 > 94 95 96 97 98 .. 99 >> Next 