Generic properties of invariant measures of full-shift systems over perfect separable metric spaces

Abstract

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, $T$, in a product space whose alphabet is a perfect and separable metric space (thus, complete and uncountable). More specifically, we show that the set of invariant measures with upper Hausdorff dimension equal to zero and lower packing dimension equal to infinity is a dense $G_\delta$ subset of $\mathcal{M}(T)$, the space of $T$-invariant measures endowed with the weak topology. We also show that the set of invariant measures with upper rate of recurrence equal to infinity and lower rate of recurrence equal to zero is a $G_\delta$ subset of $\mathcal{M}(T)$. Furthermore, we show that the set of invariant measures with upper quantitative waiting time indicator equal to infinity and lower quantitative waiting time indicator equal to zero is residual in $\mathcal{M}(T)$.