Flow-invariant hypersurfaces in semi-dispersing billiards

Details Flow-invariant hypersurfaces in semi-dispersing billiards Flow-invariant hypersurfaces in semi-dispersing billiards

2006-03-15

arxiv.org/abs/math/0603360v1

Annales Henri Poincare, Vol. 8 (2007), 475-483

Mathematics

Dynamical Systems

Scientific paper

10.1007/s00023-006-0313-5

This work results from our attempts to solve Boltzmann-Sinai's hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of $J$ grows at least linearly in time. Our proof is independent of the solution of the Boltzmann-Sinai hypothesis, and we provide a complete and self-contained argument here.

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