# American Institute of Mathematical Sciences

December  2012, 32(12): 4445-4466. doi: 10.3934/dcds.2012.32.4445

## Free path of billiards with flat points

 1 Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003

Received  March 2011 Revised  May 2012 Published  August 2012

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.
Citation: Hong-Kun Zhang. Free path of billiards with flat points. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4445-4466. doi: 10.3934/dcds.2012.32.4445
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