Journal
JOURNAL OF MODERN DYNAMICS
Volume 5, Issue 4, Pages 665-709Publisher
AMER INST MATHEMATICAL SCIENCES
DOI: 10.3934/jmd.2011.5.665
Keywords
Dispersing billiards; transfer operator; spectral gap; limit theorems
Categories
Funding
- NSF [DMS-0801139]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0901448, 0801139, 1101572] Funding Source: National Science Foundation
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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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