Abstract:
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.  For the billiard map
associated with a Lorentz gas of both finite and infinite horizon, we
construct generalized function spaces on which the transfer operator is
quasi-compact and has a spectral gap.  This framework gives a unified
approach to proving the statistical properties and various limit laws
associated with billiards, such as exponential decay of correlations,
central limit theorem and large deviation estimates.  It also has
potential applications to many classes of perturbations as well as the
billiard flow.  This is joint work Hong-Kun Zhang.