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 by Young (Ergod. Theory Dyn. Syst. 2(1):109–124, 1982) 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 pointwise satisfied, 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 (Trans. Am. Math. Soc. 190:285–299, 1974) for Lipschitz transformations which satisfy the specification property.

我们在这项工作中研究了一些可以估计或确定q广义分形维数\(D^{\pm }_{\mu }(q)\) , \(q\in { \mathbb {R}}\)，与紧凑度量空间上的连续变换相关的不变度量。特别是，我们提出了 Young 的杨氏定理的替代证明（Ergod. Theory Dyn. Syst. 2(1):109–124, 1982），用于与\(C^ {1+\alpha }\) -Axiom 二维紧致黎曼流形M 上的系统. 我们还根据度量熵提出了对 Brin-Katok 定理逐点满足的遍历度量的广义分形维数的估计。此外，对于扩展同胚（如\(C^1\) -Axiom A 系统），我们证明了一组不变测度使得\(D_\mu ^+(q)=0\) ( \(q\ge 1\) ) 在双曲线度量下是通用的（考虑到弱拓扑）。我们还表明，对于每个\(s\in [0,1)\)，\(D^{+}_{\mu }(s)\)受拓扑熵的约束，直到一个常数，也在双曲线度量下。最后，我们表明，对于某些动态系统，不变测度的度量熵通常为零，从而解决了 Sigmund（Trans. Am. Math. Soc. 190:285–299, 1974）对 Lipschitz 变换提出的猜想，该猜想满足规范属性。