Hausdorff measure of sets of distributional chaotic pairs for shift maps

Abstract

Let ${\sigma }_K:{\sum }_K\to {\sum }_K$ 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 $\epsilon \to 0^{+}$. Then ${\dim }_H{D}_{{\sigma }_K}([p,q])={\dim }_H{E}_{{\sigma }_K}([p,q])=2-q$. Moreover, ${\mathcal{H}}^{2-q}(E_{{\sigma }_K}([p,q]))=1$ when $q=0$ and ${\mathcal{H}}^{2-q}(E_{{\sigma }_K}([p,q]))=+\infty$ when $q>0$. Meanwhile, ${\mathcal{H}}^{2-q}(D_{{\sigma }_K}([p,q]))=+\infty$ when $q=1$ and ${\mathcal{H}}^{2-q}(D_{{\sigma }_K}([p,q]))=0$ when $q<1$.