Journal of Modern Dynamics (JMD)

Spectral analysis of the transfer operator for the Lorentz gas

Pages: 665 - 709, Issue 4, October 2011      doi:10.3934/jmd.2011.5.665

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Mark F. Demers - Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824, United States (email)
Hong-Kun Zhang - Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003, United States (email)

Abstract: 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:  Dispersing billiards, transfer operator, spectral gap, limit theorems.
Mathematics Subject Classification:  Primary: 37D50; Secondary: 37A25, 37C30.

Received: July 2011;      Revised: February 2012;      Available Online: March 2012.