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.
The K-mixing property is proved for the simplest, non-trivial semi-dispersing billiard: that on the 3D torus with two cylindric scatterers (systems of elastic hard spheres can be represented as higher-dimensional toric billiards with cylindric scatterers). The paper also provides o method for a stronger, topological description of a constructively defined zero-measure set of points not necessarily belonging to open ergodic components because only this set could separate the ergodic components.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.