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arXiv · 1906.07995

Uniform recurrence properties for beta-transformation

Lixuan Zheng·Min Wu
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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 $\hat{r}>1$.

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Lixuan Zheng, Min Wu. 2019-06-19. Uniform recurrence properties for beta-transformation. https://doi.org/10.1088/1361-6544/ab8a65

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