A note on the fractal dimensions of invariant measures associated with $C^{1+α}-diffeomorphisms, expanding and expansive homeomorphisms

Alexander Condori, Silas L. Carvalho

Published 2019-08-02Version 1

We show in this work that the upper and the lower generalized fractal dimensions $D^{\pm}_{\mu}(q)$, for each $q\in\mathbb{R}$, of an ergodic measure associated with an invertible bi-Lipschitz transformation over a Polish metric space are equal, respectively, to its packing and Hausdorff dimensions. This is particularly true for hyperbolic ergodic measures associated with $C^{1+\alpha}$-diffeomorphisms of smooth compact Riemannian manifolds, from which follows an extension of Young's Theorem (Young, L., S. Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems, 2(1):109-124, 1982). Analogous results are obtained for expanding systems. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures with zero correlation dimension, under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\ge 0$, $D^{+}_{\mu}(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric.