Generalized fractal dimensions of invariant measures of full-shift systems over uncountable alphabets: generic behavior

Abstract

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 $X=M^{\mathbb{Z}}$ 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.