Spectral analysis of the transfer operator for the Lorentz gas
Mark F. Demers Hong-Kun Zhang
Journal of Modern Dynamics 2011, 5(4): 665-709 doi: 10.3934/jmd.2011.5.665
We study the billiard map associated with both the finite- and infinite-horizon Lorentz gases having smooth scatterers with strictly positive curvature. We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasicompact. The mixing properties of the billiard map then imply the existence of a spectral gap and related statistical properties such as exponential decay of correlations and the Central Limit Theorem. Finer statistical properties of the map such as the identification of Ruelle resonances, large deviation estimates and an almost-sure invariance principle follow immediately once the spectral picture is established.
keywords: spectral gap limit theorems. transfer operator Dispersing billiards
Free path of billiards with flat points
Hong-Kun Zhang
Discrete & Continuous Dynamical Systems - A 2012, 32(12): 4445-4466 doi: 10.3934/dcds.2012.32.4445
In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
keywords: nonuniformly hyperbolic flat point. infinite horizon Dispersing billiards free path

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