Escape rates and physical measures for the infinite horizon Lorentz gas with holes

We study the statistical properties of the infinite horizon Lorentz gas after the introduction of small holes. Our basic approach is to prove the persistence of a spectral gap for the transfer operator associated with the billiard map in the presence of such holes. The new feature here is the interaction between the holes and the infinite horizon corridors, which causes previous approaches to fail. In order to overcome this difficulty, we redefine the Banach spaces on which we consider the action of the transfer operator. In this modified setting, we recover a complete set of results for the open system: Existence of a unified exponential rate of escape and limiting conditionally invariant measure for a large class of initial distributions, the convergence of the physical conditionally invariant measure to the smooth invariant measure for the billiard as the size of the hole tends to zero, and the characterization of the escape rate via a notion of pressure on the survivor set.