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 $\hat{r}>1$.

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β 变换的统一递归特性

对于任何 $\beta > 1$，令 $T_\beta: [0,1)\rightarrow [0,1)$ 是 $T_\beta x=\beta x \mod 1 定义的 $\beta$-transformation $. 我们研究了在 $\beta$ 变换到点本身的情况下点的轨道的均匀递归特性。获得具有规定的均匀重复率的点集的大小。更准确地说，对于任何 $0\leq \hat{r}\leq +\infty$，集合 $$\left\{x \in [0,1): \forall\ N\gg1, \exists\ 1\leq n \leq N, {\rm\ st}\ |T^n_\beta xx|\leq \beta^{-\hat{r}N}\right\}$$ 是豪斯多夫维数 $\left(\frac {1-\hat{r}}{1+\hat{r}}\right)^2$ 如果 $0\leq \hat{r}\leq 1$ 并且如果 $\hat{r}>1$ 是可数的.