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

Silas L. Carvalho, Alexander Condori

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 countable set. More specifically, we show that the set of invariant measures with infinite packing dimension equal to infinity is a dense $G_δ$ subset of $\mathcal{M}(T)$, the space of $T$-invariant measures endowed with the weak topology, where the alphabet $M$ is a countable 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 $G_δ$ subset of $\mathcal{M}(T)$, where the alphabet $M$ is a countable compact metric space. This improves the results obtained by Carvalho and Condori in \cite{AS} and \cite{AS2}, respectively. Furthermore, we discuss the dynamical consequences of such results, regarding the upper recurrence rates and upper quantitative waiting time indicator for typical orbits, and how the fractal dimensions of invariant measures and such dynamical quantities behave under an $α$-Hölder conjugation.

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

Abstract

In this work, we are interested in characterizing typical (generic) dimensional properties of invariant measures associated with the full-shift system, , in a product space whose alphabet is a countable set. More specifically, we show that the set of invariant measures with infinite packing dimension equal to infinity is a dense subset of , the space of -invariant measures endowed with the weak topology, where the alphabet is a countable Polish metric space. We also show that the set of invariant measures with upper -generalized fractal dimension (with ) equal to infinity is a dense subset of , where the alphabet is a countable compact metric space. This improves the results obtained by Carvalho and Condori in \cite{AS} and \cite{AS2}, respectively. Furthermore, we discuss the dynamical consequences of such results, regarding the upper recurrence rates and upper quantitative waiting time indicator for typical orbits, and how the fractal dimensions of invariant measures and such dynamical quantities behave under an -Hölder conjugation.
Paper Structure (6 sections, 19 theorems, 77 equations)

This paper contains 6 sections, 19 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. $PD$, $\overline{\mathcal{R}}$ and $\overline{\mathscr{R}}$ are residual sets
  5. $CD$ is a dense $G_\delta$ set for each $q>1$
  6. Lower packing, upper $q$-generalized fractal dimensions of invariant measures and upper recurrence rate, upper quantitative waiting time indicator under $\alpha$-Hölder conjugations

Key Result

Proposition 1.1

Let $\mu$ be a probability measure on $X$. Then,

Theorems & Definitions (43)

  • Definition 1.1: radius packing $\phi$-premeasure, Cutler1995
  • Definition 1.2: radius packing $\phi\textrm{-}$measure, Cutler1995
  • Definition 1.3: Hausdorff $\phi\textrm{-}$measure, Cutler1995
  • Definition 1.4: Hausdorff and packing dimensions of a set, Cutler1995
  • Definition 1.5: lower and upper packing and Hausdorff dimensions of a measure,Mattila
  • Proposition 1.1
  • Definition 1.6: Energy function
  • Theorem 1.1: Pesin Pesin1993Pesin1997
  • Definition 1.7: Generalized fractal dimensions
  • Proposition 1.2: Proposition 1.2 in AS2
  • ...and 33 more