Measures of Maximal Dimension

Abstract

We establish in this chapter the existence of ergodic measures of maximal dimension for hyperbolic sets of conformal diffeomorphisms. This is a dimensional version of the existence of ergodic measures of maximal entropy. A crucial difference is that while the entropy map is upper semicontinuous, the map ν→dim _H ν is neither upper semicontinuous nor lower semicontinuous. Our approach is based on the thermodynamic formalism. It turns out that for a generic diffeomorphism with a hyperbolic set, there exists an ergodic measure of maximal Hausdorff dimension in a particular two-parameter family of equilibrium measures. On the other hand, generically there exists no measure of full dimension, in strong contrast with what happens in the case of repellers (see Chapter 4).