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

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.