times a$ and $\times b$ empirical measures, the irregular set and entropy

• Published in: Ergodic theory and dynamical systems Vol. 44; no. 6; pp. 1673 - 1692
• Main Author: USUKI, SHUNSUKE
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2024
• Subjects: Original Article; Hausdorff dimension; entropy; multiplication on the circle; 37A35; 37E10; equidistribution; 11J71; irregular set
• ISSN: 0143-3857; 1469-4417
• DOI: 10.1017/etds.2023.60

For integers a and $b\geq 2$ , let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ . The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.