标题： 开放动力系统与移动空洞 显示英文标题

标题： Open dynamical systems with a moving hole

摘要： 给定一个整数$b\ge 3$，设$T_b: [0,1)\to [0,1); x\mapsto bx\pmod 1$是单位圆上的扩张映射。 对于任意的$m\in\mathbb{N}$和$\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N}$，令\[ K^\omega=\left\{x\in[0,1): T_b^n(x)\notin I_{\omega^n}~\forall n\geq 0\right\},\]，其中$I_{\omega^n}$是由$\omega^n$生成的$b$-进制基本区间。 然后，$K^\omega$被称为相对于孔序列的开动力系统$([0,1),T_b,I_\omega)$的幸存者集$I_\omega=\left\{I_{\omega^n}: n\geq 0\right\}$。我们证明了$K^\omega$的豪斯多夫维数和下箱维数总是相等的，并且$K^\omega$的包络维数和上箱维数也相等。此外，我们给出了$K^\omega$维数的精确上下界，这些界可以显式计算。 对于任意可容许的$\alpha\leq \beta$，存在无穷多个$\omega$使得$\dim_H K^\omega=\alpha$和$\dim_P K^\omega=\beta$。作为应用，我们研究了丢番图逼近中的 badly approximable 数。 对于任意球序列 $\left\{B_n\right\}$，令 $K\left(\left\{B_n\right\}\right)$ 为 $x\in[0,1)$ 的集合，满足 对于除有限个 $n\geq 0$ 之外的所有 $T_b^n(x)\notin B_n$。 假设$\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)$存在，我们证明$\dim_H K\left(\left\{B_n\right\}\right)=1$当且仅当$\lim_{n\to\infty}\operatorname{diam} \left(B_n\right)=0$。 对于$\mathbb{N}$上的任意正函数$\phi$，令$E\left(\phi\right)$为满足$|T_b^n (x)-x|\geq \phi(n)$对几乎所有$n$成立的$x\in[0,1)$的集合。 如果$\lim_{n\to\infty}\phi(n)$存在，则$\dim_H E(\phi)=1$当且仅当$\lim_{n\to\infty}\phi(n)=0$。 我们的结果可以应用于研究矩阵的联合谱半径。 我们证明了与关联邻接矩阵的联合谱半径相关的有限性性质成立。 显示英文摘要

摘要： 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 $\omega=\omega^0\omega^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N}$ 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.