Open dynamical systems with a moving hole

TL;DR

This work develops a comprehensive dimension theory for open dynamical systems with moving holes under the base-$b$ expanding map $T_b(x)=bx\pmod1$. By mapping survivor sets $K^{\omega}$ to a symbolic framework and utilizing graph-directed matrices $A_{\mathbf d}$, it obtains precise formulas for $\dim_H K^{\omega}$ and $\dim_P K^{\omega}$ as liminf/limsup of products of these matrices, and establishes that Hausdorff and lower box dimensions coincide, as do packing and upper box dimensions. It identifies two special ω-types—progressively overlapping (PO) and totally distinct (TD)—in which all fractal dimensions coincide and $K^{\omega}$ is $s$-Ahlfors regular for explicit Pisot-number-based exponents; it also shows dense, intermediate-dimension phenomena across the whole parameter range. An intermediate-value theorem for dimensions demonstrates the full spectrum of possible dimension pairs $(\dim_H,\dim_P)$ within a natural interval, including cases where $\dim_L<\dim_H<\dim_P<\dim_A$, and the results extend to Diophantine approximation applications and to joint spectral radius theory, including a finiteness property for the associated adjacency matrices. The paper also discusses regularity patterns, alternative constructions yielding Ahlfors-regular sets, and extensions to more general moving-hole configurations and higher-dimensional self-similar frameworks.

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 $ω=ω^0ω^1\ldots\in(\left\{0,1,\ldots,b-1\right\}^m)^\mathbb{N_0}$ let \[ K^ω=\left\{x\in[0,1): T_b^n(x)\notin I_{ω^n}~\forall n\geq 0\right\},\] where $I_{ω^n}$ is the $b$-adic basic interval generated by $ω^n$. Then $K^ω$ is called the survivor set of the open dynamical system $([0,1),T_b,I_ω)$ with respect to the sequence of holes $I_ω=\left\{I_{ω^n}: n\geq 0\right\}$. We show that the Hausdorff and lower box dimensions of $K^ω$ always conincide, and the packing and upper box dimensions of $K^ω$ also coincide. Moreover, we give sharp lower and upper bounds for the dimensions of $K^ω$, which can be calculated explicitly. For any admissible $α\leq β$ there exist infinitely many $ω$ such that $\dim_H K^ω=α$ and $\dim_P K^ω=β$. 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 $φ$ on $\mathbb{N}$, let $E\left(φ\right)$ be the set of $x\in[0,1)$ satisfying $|T_b^n (x)-x|\geq φ(n)$ for all but finitely many $n$. If $\lim_{n\to\infty}φ(n)$ exists, then $\dim_H E(φ)=1$ if and only if $\lim_{n\to\infty}φ(n)=0$. 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.

Theorems & Definitions (57)

• Theorem 1.1
• Remark 1.2
• Example 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Theorem 1.9
• Remark 1.10