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

## TL;DR

This paper investigates the typical behavior of invariant measures in full-shift dynamical systems over uncountable alphabets, revealing that most have zero generalized fractal dimensions for some parameters and infinite for others, highlighting complex measure properties.

## Contribution

It demonstrates that in full-shift systems over uncountable alphabets, a typical invariant measure exhibits zero or infinite generalized fractal dimensions depending on the parameter, extending understanding of measure complexity.

## Key findings

- Typical invariant measures have zero lower q-generalized fractal dimension for all q>0.
- In full-shift systems, typical measures have infinite upper q-correlation dimension for q>1.
- Typical measures have zero lower s-generalized and infinite upper q-generalized dimensions for certain parameters.

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