Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(=d _H–1)=lnj, which links the Hausdorff dimensiond _H of an attractor to a free energy functionalF() defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.