# An introduction to ergodic theory - Walters P.

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We also get the following corollary of Them cm 10.1 for a differentiable map T of a smooth manifold M. Recall that M(M, T) denotes the collection of all T-invariant probability measures on the ff-algebra, ЩМ), of Borel subsets of M.

Corollary 10.1.2. Let T:M M be a C1 -differentiable map of the compact manifold M and take any Riemanniun metric on M. There exists В e 06{№) with ТВ c: В and m(B) — 1 Vm e M(m, T), and a measurable function %:M -* R u { — oc} such that

lim ^ log||Tx(rn)|| = *(x) Vx e B.

n~* oo «I

We have x(x) <; sup{||r>,T'|| e M}, %lTx) = x(x) Vx e В and <

lim - Гlog||TJC(7’")|| <Jm = inf- Г1о§||тх(Т")||£/т

n-ao П •> „ П J

= J/Xх) dm Vw e M(M, T).

The set В and the function % do not depend on the Riemannian metric chosen.

Let us now turn to the problem of understanding how к^||тж(Т")г|| varies with n Tvhen v e тXM. This motivates the following result of V. I. Oseledets (Oseledets [1]).

$10.2 The SuDjddiii\e Ergodic Theorem and the Multiplicative Ergodic Theorem

233

Theorem 10.2 (Multiplicative Ergodic Theorem). Let T be measure-preserving transformation of the pmhability space (Л m). Let A: X -» L{Rk, Rk) be measurable and suppose (log||.4(.\)||)+ e L'(ni). There exists Be,J with ТВ <=■ В and in(B) = 1 with the following properties.

(a) There is a measurable function s:B —* Z+ with s ° T = s.

(b) If x e В there are real numbers л(1,(л) < /.<2)(.v) < • • • < /.(5(л,)(х), where /m(\) could be — со.

(c) If xe В there are linear subspaces {0} = K<0)(x) c= Kl,,(v) c: • • ■ c K<i(a"(.y) = R\ of Rk.

(d) if xe В and 1 < i ^ .s(x) then

1

lim - log||.4(T" 1x) * ■ ■ о A(Tx) ° Л(х)(г)||

П-* ЗГ И

= ;.("(л) far alive Vu\x)\ Vv "1 >(*).

(e) The function/}0 is defined and measurable on {x|s(x) 2: /} and /.u\Tx) — /.(1>(.v) on this set.

(f) /КлКИ"(л)) с К'-’П v) ifs[x) £ i.

Remarks

(1) For x e B, Theorem 10.2 gives the behaviour of log||,4(T"-*.v) • ■ о /4(Т.\) /1(х)1-|| for all t'G Rk and the behaviour is determined oy the s(x) numbers /."’(x),.... Aw*”(x).

(2) If we take(A’v^?,;?i) to be the trivial probability space consisting of one point and T is the identity transformation then the result reduces to saying that for a linear transformation A of Rk these are subspaces {0} = K,0) c ViUcz Vi2>c •••c= »/<'• = Rk such that Ц^'Ц1 n->• t?' if ve

where e< e*'2’ < • ■■ < eя“’ are the district numbers which are absolute values of eigenvalues of A.

(3) When к = 1. Theorem 10.2 can be stated- if g:X -* R is measurable and g+ e Ll(m) then lim„_a (l/n)X"=o ff(T'x) = gj(x) exists a.e. but could take on the value — oo.

(4) If T is ergodic then s(x) is constant a.e. and each A(,)(x) is constant a.e.

(5) The numbers /."’(v),..., лЫх))(х) are called the (Liapunov) characteristic exponents of the system (T,A) at x, and K<n(x)c= K,2)fx) a ■ • • c: l/<s(Jt|’(x) = Rk is called the associated filtration. The number w(,|(x) = dim V'Xx) — dim Vй ~ "(v) is callcd the multiplicity of /."’(x).

(6) It follows that the function *(x) occurring in corollary 10.1.1 is /}sM)(x).

(7) Because of its use by Margulis in the study of algebraic groups Raghunathan (Raghunathan [1]) gave another proof of Theorem 10.2 that is valid for local fields. A version of this proof is given in Ruelle [4].

When each /4(x) is an invertible linear transformation we have the following result. We use GL(Rk) to denote the space of invertible linear transformation of Rk.

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10 Applical.on.-. and Other Topics

Theorem 10.3. Let 7 be an invertible measure-preserving transformat ion of the probability space (X, J#,m). Let A: X -* GHRk) be и measurable function with (log||/4(\)||)+ e Ll[m) ami (log||(/l(x))- 11|)+ e Ll(m). There exists С e J# with ТС = С and n,(C) — 1 such mat for each xe С there is a direct sum decomposition of Rk into linear subspaces Rk « W{ 1 ’(v) © H/<a,( v) © • • • © <) with

lim - log||/Hrn~1v) о • • - о /1(Тх) ° Л(л)(г)|| = /."'(я)

п— x И

and

lim - logj](/4(T_1\') с • • • о /4(T-"x))_1(i’)|| = -Г'М if 0 Ф г е W(i\x).

п-* к Л

The function /.(1>(x) is never — oo, and /tfxOW'Vx) = W(,\Tx) if i < s{ c). Remarks

(1) When (А\.1в,т) consists of one point and A e GHRk) this theorem reduces to the decomposition Rk = W1' © И^2’ © ■ ■ • © W,s' into subspaces such that AWU) = Wut and all eigenvalues of A | WM have the same absolute value e*"\

(2) See Ruelle [4] for the proof.

There is the following version of the multiplicative ergodic theorem for dilleientiable maps.

Theorem 10.4. Let M be a compact C° manifold and let T'.M - M be a C1 differentiable map. Choose a Riemannian metric on M. Then there exists В e j£(M) with ТВ cz В and m(B) =1 Vw e M(M, T) with the following properties.

(a) There is a measurable function s:B -»Z+ with s° T = s.

(b) If xe В there are real numbers AU)(x) < /<2)(x) < ■ • • < /.iMx))(x\ where A(1)(\) could be — oo.

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