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

Abstract

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, [Formula: see text], in a product space whose alphabet is a perfect Polish metric space (thus, 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 [Formula: see text] subset of [Formula: see text], the space of [Formula: see text]-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 [Formula: see text] subset of [Formula: see text]. 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 [Formula: see text].