Uniform recurrence properties for beta-transformation

For any $β> 1$, let $T_β: [0,1)\rightarrow [0,1)$ be the $β$-transformation defined by $T_βx=βx \mod 1$. We study the uniform recurrence properties of the orbit of a point under the $β$-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 $\left\{x \in [0,1): \forall\ N\gg1, \exists\ 1\leq n \leq N, {\rm\ s.t.}\ |T^n_βx-x|\leq β^{-\hat{r}N}\right\}$ 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 $\hat{r}>1$.

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