A “Transversal” Fundamental Theorem for semi-dispersing billiards

Abstract

For billiards with a hyperbolic behavior, Fundamental Theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds. A "Transversal" Fundamental Theorem has recently been suggested by the present authors to prove global ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems of N ^ 3 elastic hard balls conjectured by the celebrated Boltzmann-Sίnai ergodίc hypothesis. (In fact, the suggested "Transversal" Fundamental Theorem has been successfully applied by the authors in the cases N = 3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtain local ergodicity of semi-dispersing billiards. Our theorem, however, is stronger even in their case, too, since its conditions are simpler and weaker. Moreover, a complete set of conditions is formulated under which the Fundamental Theorem and its consequences like the Zig-zag theorem are valid for general semi-dispersing billiards beyond the utmost interesting case of systems of elastic hard balls. As an application, we also give conditions for the ergodicity (and, consequently, the K-property) of dispersingbilliards. "Transversality" means the following: instead of the stable and unstable foliations occurring in the Chernov-Sinai formulation of the stable version of the Fundamental Theorem, we use the stable foliation and an arbitrary nice one transversal to the stable one.

Exponential Decay of Correlations in Multi-Dimensional Dispersing Billiards

“…Similar metric-type quantities have been used before, see e.g. the quadratic form Q used in [17], or the pseudo-metric called "p-metric" in [5,15,19]. These are known to increase on the unstable cone, but are not well-behaved on general tangent vectors.…”

“…, n. These are exactly the preimages of points that start out from some T W i,k and get close to some singularity within the first n − 1 iterates, in other words, the preimages of G i,k . This is the reason for constructing the n-gap as: 15) where the superscript F stands for future.…”

Flow-Invariant Hypersurfaces in Semi-Dispersing Billiards

“…That theorem was proven under an assumption later referred to as Chernov-Sinai Ansatz [KSS90,KSS91]; roughly speaking it states that typical points on singularity manifolds must be hyperbolic.…”

Ergodicity of two hard balls in integrable polygons

“…A billiard (on billiards in general, see subsection 2.1) is called semi-dispersing if any smooth component of the boundary ∂Q is convex as seen from the outside of Q. Semidispersing billiards are typically hyperbolic systems with singularities. We only give a short discussion of these phenomena, for a detailed exposition see [BCST] or [KSSz2]. All our arguments on semi-dispersing billiards are self contained.…”