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Escape rates and physically relevant measures for billiards with small holes. (English) Zbl 1225.37051
This paper focuses on the subject of leaky dynamical systems, where the leaks in the system are caused by possible holes in the phase space. The system starts with an initial probability distribution and as the mass leaks out of the system by time, the main concern is the distribution of the remaining mass at that instant. This is also related to the rate of escape of the mass from the system. The authors give a rigorous and detailed study of these distribution measures on billiard systems with small holes. Such measures turn out to be physically relevant and correspond to the two dimensional periodic Lorentz gas model. For these systems, they prove the existence of a common rate of escape and a common limiting distribution for a large class of initial distributions including the ones arising from the Liouville measure. In this respect, the authors construct Markov tower extensions that are compatible with holes in the system. These towers are related to sufficiently strong hyperbolic properties. The actual construction of these towers is quite elaborate and forms the heart of this paper. The information obtained is then interpreted for the billiard system, proving weak convergence of all the relevant distribution measures.

##### MSC:
 37D50 Hyperbolic systems with singularities (billiards, etc.) 37C40 Smooth ergodic theory, invariant measures 37N20 Dynamical systems in other branches of physics
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