On dichotomy law for beta-dynamical system in parameter space

Abstract

Let $\varphi :\mathbb {N}\rightarrow (0,1]$ be a positive function and $T_{\beta }$ be the beta-transformation for any $\beta >1$. We prove that the set

$$\begin{aligned} E(0, \varphi )=\{\beta >1:|T^{n}_{\beta }1-0|<\varphi (n) \text { for infinitely many } n\in \mathbb {N}\} \end{aligned}$$

is of zero or full Lebesgue measure in $(1,+\infty )$ according to $\sum \varphi (n)<+\infty$ or not. As an application, we determine the exact Lebesgue measure of the set

$$\begin{aligned} \mathfrak {E}(0, \{l_{n}\})=\{\beta >1:|T^{n}_{\beta }1-0|<\beta ^{-l_{n}} \text { for infinitely many } n\in \mathbb {N}\}, \end{aligned}$$

where $\{l_{n}\}_{n\ge 1}$ is a sequence of non-negative real numbers.