Dimension of generic self-affine sets with holes

Let (, s) be a dynamical system, and let U. . Consider the survivor set U = x. | s n (x) /. U for all n of points that never enter the subset U. We study the size of this set in the case when is the symbolic space associated to a self-affine set , calculating the dimension of the projection of U as a subset of and finding an asymptotic formula for the dimension in terms of the Kaenmaki 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 are diagonal; and when they are such that the pressure is differentiable at its zero point, and the Kaenmaki measure is a strong-Gibbs measure.