Fractal Dimensions for Repellers of Maps with Holes

Abstract

In this work we study the Hausdorff dimension and limit capacity for repellers of certain non-uniformly expanding maps f defined on a subset of a manifold. This subset is covered by a finite number of compact domains with pairwise disjoint interiors (the complement of the union of these domains is called hole) each of which is mapped smoothly to the union of some of the domains with a subset of the hole. The maps are not assumed to be hyperbolic nor conformal. We provide conditions to ensure that the limit capacity of the repeller is less than the dimension of the ambient manifold. We also prove continuity of these fractal invariants when the volume of the hole tends to zero.