Nandor Simanyi

A “transversal” fundamental theorem for semi-dispersing billiards

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 proveglobal ergodicity (and then, as an easy consequence, the K-property) of semidispersing billiards, in particular, the global ergodicity of systems ofN≧3 elastic hard balls conjectured by the celebratedBoltzmann-Sinai ergodic hypothesis. (In fact, the suggested “Transversal” Fundamental Theorem has been successfully applied by the authors in the casesN=3 and 4.) The theorem generalizes the Fundamental Theorem of Chernov and Sinai that was really the fundamental tool to obtainlocal 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 dispersing-billiards. “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-property of three billiard balls

Sinais strengthened version of the ergodic hypothesis is proved for three billiard balls on the v-dimensional torus: On connected components of the submanifold of the phase space specified by the trivial conservation laws of the energy and of the trajectory of the center of mass, the system is a K-flow. To cope with the difficulty that in the isomorphic one-particle-billiard system the scatterers are not strictly convex, geometric-algebraic, ergodic-theoretic and topological methods are elaborated

Dual polygonal billiards and necklace dynamics

We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.

The

A further step is achieved toward establishing the celebrated Boltzmann-Sinai ergodic hypothesis: for systems of four hard balls on the ν-torus (ν>2) it is shown that, on the submanifold of the phase specified by the trivial conservation laws, the system is aK-flow. All parts of our previous demonstration providing the analogous result for three hard balls are simplified and strengthened. The main novelties are: (i) A refinement of the geometric-algebraic methods used earlier helps us to bound the codimension of the arising implicitly given set of degeneracies even if we can not calculate their exact dimension that was possible for three-billiards. As a matter of fact, it is this part of our arguments, where further understanding and new ideas are necessary before attacking the general ergodic problem; (ii) In the “pasting” part of the proof, which is a sophisticated version of Hopfs classical device, the arguments are so general that it is hoped they work in the general case, too. This is achieved for four balls, in particular, by a version of the Transversal Fundamental Theorem which, on one hand, is simpler and more suitable for applications than the previous one and, on the other hand, as we have discovered earlier, is the main tool to prove global ergodicity of semi-dispersing billiards; (iii) The verification of the Chernov-Sinai ansatz is essentially simplified and the new idea of the proof also promises to work in the general case.

K

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). They also provide a 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.

-property of four billiard balls

We consider the system of N (≥2) hard disks of masses m1,...,mN and radius r in the flat unit torus 𝕋2. We prove the ergodicity (actually, the B-mixing property) of such systems for almost every selection (m1,...,mN;r) of the outer geometric parameters.

Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus

Abstract. We consider the system of

Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

N (\geq 2)