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

TL;DR

This paper investigates the behavior of empirical measures under the $ imes a, imes b$ action on the circle, revealing that divergence is prevalent while convergence to nontrivial invariant measures is extremely rare.

Contribution

It establishes the Hausdorff dimension of sets of points with divergent empirical measures and those approaching nontrivial invariant measures under $ imes a, imes b$ dynamics.

Findings

Set of points with non-converging empirical measures has Hausdorff dimension 1.

Set of points whose empirical measures approach a nontrivial invariant measure has Hausdorff dimension zero.

Provides equidistribution results outside a set of Hausdorff dimension zero.

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.