Hausdorff measure of sets of distributional chaotic pairs for shift maps

TL;DR

This paper investigates the Hausdorff measure and dimension of sets of pairs with specific distributional chaotic properties under shift maps, revealing precise measure and dimension results depending on parameters.

Contribution

It provides exact Hausdorff measure and dimension formulas for sets of distributionally chaotic pairs in shift maps, extending understanding of their fractal structure.

Findings

Hausdorff dimension of sets equals 2-q

Hausdorff measure is 1 or infinite at q=0, and zero or infinite at q=1

Explicit measure and dimension results for distributional chaos sets

Abstract

Let $\sigma_{K}: \sum_{K}\rightarrow \sum_{K}$ be a shift map. For an interval $[p,q]\subset[0,1]$, let $D_{\sigma_{K}}([p,q])$ denote the set of pairs for which the density spectrum of the $\epsilon$-approach time set equals $[p,q]$ when $\epsilon$ is small and $E_{\sigma_{K}}([p,q])$ the set of pairs for which the density spectrum of the $\epsilon$-approach time set converges to $[p,q]$ when $\epsilon\rightarrow 0^+$. Then $\dim_{H} D_{\sigma_{K}}([p,q])=\dim_{H} E_{\sigma_{K}}([p,q])=2-q$. Moreover, $\mathscr{H}^{2-q}(E_{\sigma_{K}}([p,q]))=1$ when $q=0$ and $\mathscr{H}^{2-q}(E_{\sigma_{K}}([p,q]))=+\infty$ when $q>0$. Meanwhile, $\mathscr{H}^{2-q}(D_{\sigma_{K}}([p,q]))=+\infty$ when $q=1$ and $\mathscr{H}^{2-q}(D_{\sigma_{K}}([p,q]))=0$ when $q<1$.