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

Abstract

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, whether there exists a nontrivial $\times a,\times b$ invariant and ergodic measure is not known. 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 $1$ and the set of $x$ such that the empirical measures can approach a nontrivial $\times a,\times b$ 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.