Dimension of generic self-affine sets with holes

Let $(\mathrm{\Sigma },\sigma )$ be a dynamical system, and let $U\subset \mathrm{\Sigma }$. Consider the survivor set $${\mathrm{\Sigma }}_U=\{x\in \mathrm{\Sigma }\mid {\sigma }^n(x)\mathrm{\notin }U\text{for all}n\}$$ of points that never enter the subset $U$. We study the size of this set in the case when $\mathrm{\Sigma }$ is the symbolic space associated to a self-affine set $\mathrm{\Lambda }$, calculating the dimension of the projection of ${\mathrm{\Sigma }}_U$ as a subset of $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ 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.