In this paper we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each $q>0$, zero lower $q$-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system $(X,T)$ (where $X= M^{\Z}$ is endowed with a sub-exponential metric and the alphabet $M$ is a perfect and compact metric space), for which we show that a typical invariant measure has, for each $q>1$, infinite upper $q$-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each $s\in(0,1)$ and each $q>1$, zero lower $s$-generalized and infinite upper $q$-generalized dimensions.

中文翻译：

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紧凑和完美空间上全移系统不变测度的广义分形维数：通用行为

在本文中，我们表明，对于具有密集集（在弱拓扑中）周期性测度的拓扑动力系统，典型的（Baire 意义上的）不变测度对于每个 $q>0$，零下 $q$-广义分形维数。这特别意味着典型的不变度量具有零上豪斯多夫维度和零下复发率。特别有趣的是全班次系统 $(X,T)$（其中 $X= M^{\Z}$ 被赋予了一个次指数度量，字母表 $M$ 是一个完美而紧凑的度量空间） ，对此我们表明，对于每个 $q>1$，一个典型的不变度量具有无限的上 $q$-相关维度。在相同条件下，我们证明了一个典型的不变测度，对于每个 $s\in(0,1)$ 和每个 $q>1$，零下 $s$-广义和无限上 $q$-广义维度.