CENTRAL LIMIT THEOREM FOR DIMENSION OF GIBBS MEASURES IN HYPERBOLIC DYNAMICS

Abstract

For an equilibrium measure of a Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero. A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius ε chosen at random have a measure smaller (or larger) than εδ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure. Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.