Hausdorr Dimension, Strong Hyperbolicity and Complex Dynamics X0. Introduction

Let X be a compact metric space and assume that f : X ! X is a continuous map. Denote by the nonwandering set of f. An interesting and a nontrivial invariant of f is HD(()-the Hausdorr dimension of. It is usually a highly nontrivial problem to nd HD((). The seminal work of Bowen Bow2] gives HD(() as the solution to P(tt) = 0 for some special expanding maps. Here P(g) denotes the topological pressure. See also Rue2] and the recent works Bar] and Fri2]. Denote by E the set of all f-invariant ergodic probability Borel measures on M. Let HD(); 2 E be the Hausdorr dimension of HD() = inf Y;;(Y)=1 HD(Y): It is known that HD() is easy to compute in many general cases. See for example Man], You], L-Y] and Fri, 1-2]. In the above references HD() is given in terms of entropy of f (along a foliation) and the Lyapunov exponents. As the support of lies in it follows that HD(() HD(). Hence HD(() sup 2E HD(); : (0:1) In fact in the examples studied in Bow2] and Rue2] one has the equality in (0.1). In these cases HD(() = HD() and is a unique Gibbs measure given by thermodynamics formalism which is equivalent (absolutely continuous) to the Hausdorr measure on. See also Fri2]. In general a strict inequality holds in (0.1). To motivate our results consider the following example.