Abstract
We study the billiard map associated with both the finite- andinfinite-horizon Lorentz gases having smooth scatterers with strictlypositive curvature. We introduce generalized function spaces (Banach spacesof distributions) on which the transfer operator is quasicompact. Themixing properties of the billiard map then imply the existence of aspectral gap and related statistical properties such as exponential decayof correlations and the Central Limit Theorem. Finer statistical propertiesof the map such as the identification of Ruelle resonances, large deviationestimates and an almost-sure invariance principle follow immediately oncethe spectral picture is established.
Highlights
Much attention has been given in recent years to developing a framework to study directly the transfer operator associated with hyperbolic maps on an appropriate Banach space
We introduce generalized function spaces (Banach spaces of distributions) on which the transfer operator is quasi-compact
2.1 Billiard maps associated with a Lorentz gas
Summary
Much attention has been given in recent years to developing a framework to study directly the transfer operator associated with hyperbolic maps on an appropriate Banach space The goal of such a functional analytic approach is first to use the smoothing properties of the transfer operator to prove its quasi-compactness and to derive statistical information about the map from the peripheral spectrum. Two crucial assumptions in the treatment of the piecewise hyperbolic case in two dimensions have been: (1) the map has a finite number of singularity curves and (2) the map admits a smooth extension up to the closure of each of its domains of definition These assumptions and other technical difficulties have far prevented this approach from being successfully carried out for dispersing billiards.
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