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

Silas Luiz Carvalho, Alexander Condori

Published 2019-03-07Version 1

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 uncountable. %fixed dynamic system and with certain dimensional properties. In this search, 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 also 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 $\mathcal{M}(T)$.