Typicality of recurrence for Lorentz gases

Details Typicality of recurrence for Lorentz gases Typicality of recurrence for Lorentz gases

2004-10-15

arxiv.org/abs/math/0410355v2

Ergodic Theory Dynam. Systems 26 (2006), no. 3, 799-820

Mathematics

Dynamical Systems

22 pages, 5 figures

Scientific paper

10.1017/S0143385706000022

It is a safe conjecture that most (not necessarily periodic) two-dimensional Lorentz gases with finite horizon are recurrent. Here we formalize this conjecture by means of a stochastic ensemble of Lorentz gases, in which i.i.d. random scatterers are placed in each cell of a co-compact lattice in the plane. We prove that the typical Lorentz gas, in the sense of Baire, is recurrent, and give results in the direction of showing that recurrence is an almost sure property (including a zero-one law that holds in every dimension). A few toy models illustrate the extent of these results.

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