Generic dimensional and dynamical properties of invariant measures of full-shift systems over countably infinite alphabets

In this work, we show that the set of invariant measures with packing dimension equal to infinity is a dense Gδ subset of M(T), the space of T-invariant measures endowed with the weak topology, where T is the full-shift system in a product space whose alphabet M is any infinite Polish metric space. We also show that the set of invariant measures with upper q-generalized fractal dimension (with q>1) equal to infinity is a dense Gδ subset of M(T), in case M is any countably infinite compact metric space. This improves the results obtained by Carvalho and Condori in [Generic properties of invariant measures of full-shift systems over perfect polish metric spaces, Stochast. Dyn. 21(07) (2021), p. 2150040] and [Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: generic behaviour, Forum Math. 33(2) (2021), pp. 435–450], respectively. Furthermore, we obtain results regarding the upper recurrence rates and upper quantitative waiting time indicator for typical T-orbits, and how the fractal dimensions of invariant measures and such dynamical quantities behave under an α-Hölder conjugation.