Escape Rates and Physically Relevant Measures for Billiards with Small Holes

Details Escape Rates and Physically Relevant Measures for Billiards with Small Holes Escape Rates and Physically Relevant Measures for Billiards with Small Holes

2009-04-07

arxiv.org/abs/0904.1135v1

Communications in Mathematical Physics 294 (2010), no. 2, 353-388

Mathematics

Dynamical Systems

39 pages, 4 figures

Scientific paper

We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.

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