Open dynamical systems with a moving hole

Abstract

Given an integer $b\ge 3$, let $T_b: [0,1)\to [0,1); x\mapsto bx\pmod 1$ be the expanding map on the unit circle. For any $m\in\mathbb{N}$ and $\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N_0}$ let \[ K^\omega=\left\{x\in[0,1): T_b^n(x)\notin I_{\omega^n}~\forall n\geq 0\right\},\] where $I_{\omega^n}$ is the $b$-adic basic interval generated by $\omega^n$. Then $K^\omega$ is called the survivor set of the open dynamical system $([0,1),T_b,I_\omega)$ with respect to the sequence of holes $I_\omega=\left\{I_{\omega^n}: n\geq 0\right\}$. We show that the Hausdorff and lower box dimensions of $K^\omega$ always conincide, and the packing and upper box dimensions of $K^\omega$ also coincide. Moreover, we give sharp lower and upper bounds for the dimensions of $K^\omega$, which can be calculated explicitly. For any admissible $\alpha\leq \beta$ there exist infinitely many $\omega$ such that $\dim_H K^\omega=\alpha$ and $\dim_P K^\omega=\beta$. 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 $n\geq 0$. Assuming $\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)$ exists, we show that $\dim_H K\left(\left\{B_n\right\}\right)=1$ if and only if $\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)=0$. For any positive function $\phi$ on $\mathbb{N}$, let $E\left(\phi\right)$ be the set of $x\in[0,1)$ satisfying $|T_b^n (x)-x|\geq \phi(n)$ for all but finitely many $n$. If $\lim_{n\to\infty}\phi(n)$ exists, then $\dim_H E(\phi)=1$ if and only if $\lim_{n\to\infty}\phi(n)=0$. 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.