A note on the relation between the metric entropy and the generalized fractal dimensions of invariant measures

Abstract: We investigate in this work some situations where it is possible to estimate or determine the upper and the lower $q$-generalized fractal dimensions $D^{\pm}_{\mu}(q)$, $q\in\mathbb{R}$, of invariant measures associated with continuous transformations over compact metric spaces. In particular, we present an alternative proof of Young's Theorem~\cite{Young} for the generalized fractal dimensions of the Bowen-Margulis measure associated with a $C^{1+\alpha}$-Axiom A system over a two-dimensional compact Riemannian manifold $M$. We also present estimates for the generalized fractal dimensions of an ergodic measure for which Brin-Katok's Theorem is satisfied punctually, in terms of its metric entropy. Furthermore, for expansive homeomorphisms (like $C^1$-Axiom A systems), we show that the set of invariant measures such that $D_\mu^+(q)=0$ ($q\ge 1$), under a hyperbolic metric, is generic (taking into account the weak topology). We also show that for each $s\in [0,1)$, $D^{+}_{\mu}(s)$ is bounded above, up to a constant, by the topological entropy, also under a hyperbolic metric. Finally, we show that, for some dynamical systems, the metric entropy of an invariant measure is typically zero, settling a conjecture posed by Sigmund in~\cite{Sigmund1974} for Lipschitz transformations which satisfy the specification property.