Open dynamical systems with a moving hole

Abstract

Given an integer $b\ge 3$, let $T_b:[0,1)\to [0,1);x\mapsto bx(\mathrm{mod}1)$ be the expanding map on the unit circle. For any $m\in \mathbb{N}$ and $\ddot{I}‰=\ddot{I}{‰}^0\ddot{I}{‰}^1\dots \in ({\left\{0,1,\dots ,b-1\right\}}^m{)}^{{\mathbb{N}}_0}$ let \[ K^ω=\left\{x\in[0,1): T_b^n(x)\notin I_{ω^n}~\forall n\geq 0\right\},\] where $I_{\ddot{I}{‰}^n}$ is the $b$-adic basic interval generated by $\ddot{I}{‰}^n$. Then $K^{\ddot{I}}‰$ is called the survivor set of the open dynamical system $([0,1),T_b,I_{\ddot{I}}‰)$ with respect to the sequence of holes $I_{\ddot{I}}‰=\left\{I_{\ddot{I}{‰}^n}:n\ge 0\right\}$. We show that the Hausdorff and lower box dimensions of $K^{\ddot{I}}‰$ always conincide, and the packing and upper box dimensions of $K^{\ddot{I}}‰$ also coincide. Moreover, we give sharp lower and upper bounds for the dimensions of $K^{\ddot{I}}‰$, which can be calculated explicitly. For any admissible $\hat{I}\pm \le {\hat{I}}^2$ there exist infinitely many $\ddot{I}‰$ such that ${\dim }_H{K}^{\ddot{I}}‰=\hat{I}\pm$ and ${\dim }_P{K}^{\ddot{I}}‰={\hat{I}}^2$. As applications we study badly approximable numbers in Diophantine approximation. For an arbitrary sequence of balls $\left\{B_n\right\}$, let $K\left(\left\{B_n\right\}\right)$ be the set of $x\in [0,1)$ such that $T_b^n(x)\notin B_n$ for all but finitely many $n\ge 0$. Assuming ${\lim }_{n\to \infty }\mathrm{diam}\left(B_n\right)$ exists, we show that ${\dim }_{H}K\left(\left\{B_n\right\}\right)=1$ if and only if ${\lim }_{n\to \infty }\mathrm{diam}\left(B_n\right)=0$. For any positive function $\ddot{I}•$ on $\mathbb{N}$, let $E\left(\ddot{I}•\right)$ be the set of $x\in [0,1)$ satisfying $|T_b^n(x)-x|\ge \ddot{I}•(n)$ for all but finitely many $n$. If ${\lim }_{n\to \infty }\ddot{I}•(n)$ exists, then ${\dim }_{H}E(\ddot{I}•)=1$ if and only if ${\lim }_{n\to \infty }\ddot{I}•(n)=0$. Our results can be applied to study joint spectral radius of matrices. We show that the finiteness property for the joint spectral radius of associated adjacency matrices holds true.