Spectral analysis of the transfer operator for the Lorentz gas
Mark F. Demers Hong-Kun Zhang
Journal of Modern Dynamics 2011, 5(4): 665-709 doi: 10.3934/jmd.2011.5.665
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: spectral gap limit theorems. transfer operator Dispersing billiards
Limiting distributions for countable state topological Markov chains with holes
Mark F. Demers Christopher J. Ianzano Philip Mayer Peter Morfe Elizabeth C. Yoo
Discrete & Continuous Dynamical Systems - A 2017, 37(1): 105-130 doi: 10.3934/dcds.2017005

We study the dynamics of countable state topological Markov chains with holes, where the hole is a countable union of 1-cylinders. For a large class of positive recurrent potentials and under natural assumptions on the surviving dynamics, we prove the existence of a limiting conditionally invariant distribution, which is the unique limit of regular densities under the renormalized dynamics conditioned on non-escape. We also prove the existence of a Gibbs measure on the survivor set, the set of points that never enter the hole, which is an equilibrium measure for the punctured potential of the open system. We prove that the Gurevic pressure on the survivor set equals the exponential escape rate from the open system. These results extend to the non-compact setting results previously available for finite state topological Markov chains.

keywords: Open systems transfer operator conditional limiting distribution pressure variational principle

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