Dimension of generic self-affine sets with holes

TL;DR

This paper investigates the dimension of survivor sets in self-affine fractals with holes, providing formulas for how the dimension changes as the hole size shrinks, under specific matrix and measure conditions.

Contribution

It extends the understanding of the dimension of survivor sets in self-affine sets with holes, especially in cases with diagonal matrices or differentiable pressure and strong-Gibbs measures.

Findings

Derived an asymptotic formula for the dimension of survivor sets as holes shrink.

Established results for diagonal matrices and for cases with differentiable pressure and strong-Gibbs measures.

Provided explicit dimension estimates for self-affine sets with holes.

Abstract

Let $(\Sigma, \sigma)$ be a dynamical system, and let $U\subset \Sigma$. Consider the survivor set \[ \Sigma_U=\{x\in \Sigma\mid \sigma^n(x)\notin U\textrm{for all}n\} \] of points that never enter the subset $U$. We study the size of this set in the case when $\Sigma$ is the symbolic space associated to a self-affine set $\Lambda$, calculating the dimension of the projection of $\Sigma_U$ as a subset of $\Lambda$ and finding an asymptotic formula for the dimension in terms of the K\"aenm\"aki measure of the hole as the hole shrinks to a point. Our results hold when the set $U$ is a cylinder set in two cases: when the matrices defining $\Lambda$ are diagonal, and when they are such that the pressure is differentiable at its zero point, and the K\"aenm\"aki measure is a strong-Gibbs measure.