# Uniform recurrence properties for beta-transformation

## TL;DR

This paper investigates the uniform recurrence properties of points under the beta-transformation, characterizing the Hausdorff dimension of sets with prescribed recurrence rates and revealing a phase transition at a critical rate.

## Contribution

It provides a precise Hausdorff dimension formula for sets of points with specified uniform recurrence rates in beta-transformations, extending understanding of recurrence behavior.

## Key findings

- Hausdorff dimension of recurrence sets for 0 ≤ r ≤ 1 is ((1 - r)/(1 + r))^2
- Recurrence sets with rate r > 1 are countable
- The dimension formula reveals a phase transition at r=1

## Abstract

For any $\beta > 1$, let $T_\beta: [0,1)\rightarrow [0,1)$ be the $\beta$-transformation defined by $T_\beta x=\beta x \mod 1$. We study the uniform recurrence properties of the orbit of a point under the $\beta$-transformation to the point itself. The size of the set of points with prescribed uniform recurrence rate is obtained. More precisely, for any $0\leq \hat{r}\leq +\infty$, the set $$\left\{x \in [0,1): \forall\ N\gg1, \exists\ 1\leq n \leq N, {\rm\ s.t.}\ |T^n_\beta x-x|\leq \beta^{-\hat{r}N}\right\}$$ is of Hausdorff dimension $\left(\frac{1-\hat{r}}{1+\hat{r}}\right)^2$ if $0\leq \hat{r}\leq 1$ and is countable if $\hat{r}>1$.