Decay of correlations for billiards with flat points I: channel effects. (English) Zbl 1398.37015
Blokh, Alexander M. (ed.) et al., Dynamical systems, ergodic theory, and probability. In memory of Kolya Chernov. Conference dedicated to the memory of Nikolai Chernov, University of Alabama at Birmingham, Birmingham, AL, USA, May 18–20, 2015. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2773-3/pbk; 978-1-4704-4224-8/ebook). Contemporary Mathematics 698, 239-286 (2017).
Summary: In this paper we construct a special family of semidispersing billiards bounded on a rectangle with a dispersing scatterer. The billiard induces a Lorenz gas with infinite horizon when one replaces the rectangle with a torus. We assume there exist two pairs of flat points (with zero curvature) on the boundary of this scatterer, whose tangent lines form two channels in the billiard table that are each perpendicular to two parallel sides of the rectangle. We study the mixing rates of a one-parameter family of the semi-dispersing billiards, as well as those for the corresponding Lorenz gas on a torus. We show that the correlation functions of both maps decay polynomially.
For the entire collection see [Zbl 1376.37001].
For the entire collection see [Zbl 1376.37001].
MSC:
| 37A60 | Dynamical aspects of statistical mechanics |
| 37A25 | Ergodicity, mixing, rates of mixing |
| 37D50 | Hyperbolic systems with singularities (billiards, etc.) (MSC2010) |
| 37E30 | Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces |
| 60F05 | Central limit and other weak theorems |
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