Measures of maximal dimension for hyperbolic diffeomorphisms
Let \(f: M\to M\) be a \(C^{1+\varepsilon}\) diffeomorphism and \(\Lambda\subset M\) a compact locally maximal hyperbolic set. It is well known that there is a unique measure of maximal entropy on \(\Lambda\). Moreover, the function \(\nu\mapsto h_{\nu}(f)\) defined on \(f\)-invariant probability measures is upper semicontinuous and its maximum satisfies the variational principle \(\max\{h_{\nu}(f)\}=h_{\text{top}}(f)\). The authors prove that if \(\dim M=2\) and \(f|_{\Lambda}\) is topologically mixing, then there is an ergodic measure \(\mu\) of maximal (Hausdorff) dimension \(\dim_H(\mu)=\max\{\dim_H(\nu)\}\). The map \(\nu\mapsto\dim_H(\mu)\) is neither upper nor lower semicontinuous. The ergodic measure of maximal dimension is not always unique, but the authors identify classes of diffeomorphisms for which the number of such measures is finite. The proofs are based on thermodynamic formalism.
scientific article; zbMATH DE number 1969274