Sharp Error Term in Local Limit Theorems and Mixing for Lorentz Gases with Infinite Horizon

Communications in Mathematical Physics, vol. 382, pp. 1625–1689

Abstract

Abstract We obtain sharp error rates in the local limit theorem for the Sinai billiard map (one and two dimensional) with infinite horizon. This result allows us to further obtain higher order terms and thus, sharp mixing rates in the speed of mixing of dynamically Hölder observables for the planar and tubular infinite horizon Lorentz gases in the map (discrete time) case. We also obtain an asymptotic estimate for the tail probability of the first return time to the initial cell. In the process, we study families of transfer operators for infinite horizon Sinai billiards perturbed with the free flight function and obtain higher order expansions for the associated families of eigenvalues and eigenprojectors.

Authors 2

1. Françoise Pène

   Centre National de la Recherche Scientifique · Université de Bretagne Occidentale · Université de Bretagne Occidentale · Laboratoire de Mathématiques de Bretagne Atlantique

   Affiliation as printed

   Univ Brest, Université de Brest, LMBA, UMR CNRS 6205, 6 Avenue Le Gorgeu, 29238, Brest cedex, France

   Laboratoire de Mathématiques de Bretagne Atlantique

   Université de Brest

2. Dalia Terhesiu corresponding Aachen Mathematisch Instituut

   Leiden University

   Affiliation as printed

   Mathematisch Instituut, University of Leiden, Niels Bohrweg 1, 2333, Leiden, CA, The Netherlands