Discrete Lyapunov exponents and Hausdor dimension

A major concept in diierentiable dynamics is the Lyapunov exponents of a given map f. It combines the results of ergodic theory with diierential properties of f. Consider the following two closely related examples which motivate our paper. Let M be a compact surface and f : M ! M a smooth diieomorphism. Let E denote the set of Borel f-invariant ergodic measures on M. Assume that 2 E. Let h() be the-entropy of f and 1 () 2 () be the Lyapunov exponents of f. Suppose that h() > 0. The well known result of L.S. Young You] yields that h() 1() is the Hausdorr dimension of unstable manifold W u () associated with 1 (). Suppose furthermore that f is an Axiom A diieomorphism. Then for each x in the nonwandering set (f) one has the unstable manifold W u (x). The result of McCluskey-Manning M-M] yields that the Hausdorr dimension of any W u (x) \ (f) is equal to sup 2E h() 1(). The supremum is achieved for a unique Gibbs measure. Let f : CP ! CP be a rational map of the Riemann sphere CP of degree at least two. Denote by J(f) the Julia set of f. Let E be all f-invariant ergodic measures supported on J(f). For 2 E, f has two equal Lyapunov exponents 1 () = 2 () 0. Assume that h() > 0. Then the-Hausdorr dimension of J(f) (which is inf XJ(f);;(X)=1 dim H X) is equal to h() (). Suppose furhtermore that f is hyperbolic. That is, j(f m) 0 (z)j > 1; z 2 J(f), for some integer m 1. Then dim H J(f) = sup 2E h() (m). The above supremum is achieved for a unique Gibbs measure which is equivalent to the Hausdorr measure on J(f) Rue]. In these two examples the proof of the variational formula for the Hausdorr dimension is based on the notion of the topological pressure and the Bowen equation Bow1-2]. In this paper we generalize these results to a discrete setting as follows. Let < n >= f1; :::; ng be an alphabet on n symbols. Denote by N; < n > N the set of natural numbers and the space of all innnite sequences on < n > symbols equipped with the Tychonoo topology respectively. Let :< n > N !< n > N be the (one sided) shift …