Hausdorff measure of sets of distributional chaotic pairs for shift maps

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 $q<1$.