Spectral analysis of the transfer operator for the Lorentz gas

Abstract

We study the billiard map associated with both the finite- andinfinite-horizon Lorentz gases having smooth scatterers with strictlypositive curvature. We introduce generalized function spaces (Banach spacesof distributions) on which the transfer operator is quasicompact. Themixing properties of the billiard map then imply the existence of aspectral gap and related statistical properties such as exponential decayof correlations and the Central Limit Theorem. Finer statistical propertiesof the map such as the identification of Ruelle resonances, large deviationestimates and an almost-sure invariance principle follow immediately oncethe spectral picture is established.