Spectral analysis of the transfer operator for the Lorentz gas

Mark F. Demers; Hong-Kun Zhang

doi:10.3934/jmd.2011.5.665

Article Contents

2011, Volume 5, Issue 4: 665-709. Doi: 10.3934/jmd.2011.5.665

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Spectral analysis of the transfer operator for the Lorentz gas

Mark F. Demers^1, and
Hong-Kun Zhang^2,

1.
Department of Mathematics and Computer Science, Fairfield University, Fairfield CT 06824
2.
Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003

Received: June 30, 2011

Revised: January 31, 2012

Published: February 2012

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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.

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Mathematics Subject Classification: Primary: 37D50; Secondary: 37A25, 37C30.

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