Dimension of self-affine sets with holes

TL;DR

This paper analyzes the dimensions of certain non-conformal fractal sets with holes, showing how escape rates influence their box and Hausdorff dimensions.

Contribution

It provides explicit formulas for the box and Hausdorff dimensions of holes in Bedford-McMullen sets under dynamical systems.

Findings

Box dimension relates to escape rate of maximal entropy measure.

Hausdorff dimension depends on escape rate of measure of maximal dimension.

Dimensions are computed for both fixed Markov holes and shrinking metric balls.

Abstract

In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.