Uniform recurrence properties for beta-transformation

Abstract

For any $\beta >1$, let $T_{\beta }:[0,1)\to [0,1)$ be the $\beta$-transformation defined by $T_{\beta }x=\beta x\mathrm{mod}1$. 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 $0\le \hat{r}\le +\infty$, the set $$\left\{x\in [0,1):\forall N\gg 1,\exists 1\le n\le N,s.t.|T_{\beta }^{n}x-x|\le {\beta }^{-\hat{r}N}\right\}$$ is of Hausdorff dimension ${\left(\frac{1-\hat{r}}{1+\hat{r}}\right)}^2$ if $0\le \hat{r}\le 1$ and is countable if $\hat{r}>1$.

19 pages, 2 thoerems