1. Jacobi operators the definition of ^(z), 7±(^) is indeed independent of no- Moreover, 7 ± (z) 0 if a(n) is bounded. In fact, since 7±( z) 0 would imply limj-^±00 ||$(z, rij, no)|| = 0 for some subsequence rij contradicting (1.33). If 7±(2 ) = ^ ( z ) we will omit the bars. A number A G M is said to be hyperbolic at ztoo if 7±(A) T^A) 0, respectively. The set of all hyperbolic numbers is denoted by Hyp±(l). For A G Hyp±(£) one has existence of corresponding stable and unstable manifolds V±(X). Lemma 1.1. Suppose that \a(n)\ does not grow or decrease exponentially and that \b(n)\ does not grow exponentially, that is, (1.37) lim -i-ln|a(n)| = 0 , lim - ^ ln(l + |6(n)|) = 0. n—±oo 172,J n - ^ ± o o | n | If A G Hyp± (!), then there exist one-dimensional linear subspaces V±(X) C E2 such that veV±{\)& lim -^hi\\^{X,n)v\\ = --^±(X), n—±oo | n | (1.38) v^V±{\)^ lim ^1\n\\§{\,n)v\\=1±{\), respectively. Proof. Set (1.39) A(n) = and abbreviate U(z, n) = A(n)U(z, n)A(n - l)~l (1.40) l(z, n) = A(ra)*(*, n)A(O)"1. Then (1.28) translates into (1.41) u(n + 1) = U(z, n + l)w(n), £(n - 1) = {/(z, n)_1M(n), where u = Alt = (w, a?x+), and we have (1.42) detU(z,n) = l and lim T ^ m||£/(z,n)|| = 0 n-+oc \n\ due to our assumption (1.37). Moreover, min(l, a(n)) ||$(z,n)H max(l, q(n)) 1 ' } max(l,a(0)) " ||*(z,n)|| ~ min(l,a(0)) and hence linin-^ioo \n\~l In ||$(z, n)|| = \\mn^±OG |n| _ 1 In ||$(z,n)|| whenever one of the limits exists. The same is true for the limits of |n| _ 1 In ||I(z,n)i|| and \n\~x In ||$(z,n)v||. Hence it suffices to prove the result for matrices £ satisfying (1.42). But this is precisely the (deterministic) multiplicative ergodic theorem of Osceledec (see [201]). 1 0 0 a(n) 1 / 0 1 a ( n - l ) V - a ( n - l ) 2 z-b{n)
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