Dimension of generic 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.