Domokos Szász

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

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

An overview of the history of Ludwig Boltzmann’s more than one hundred year old ergodic hypothesis is given. The existing main results, the majority of which is connected with the theory of billiards, are surveyed, and some perspectives of the theory and interesting and realistic problems are also mentioned.

-property of four billiard balls

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.

Boltzmann’s Ergodic Hypothesis, a Conjecture for Centuries?

Between two absorbing barriers consider a random walk with a finite number of internal degrees of freedom and with zero drift. By using a functional-analytic approach based on the spectral theory of matrix polynomials, the asymptotics of the first-hitting probabilities is obtained when the distance of the barriers tends to infinity.

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

First return and first hitting times, local times and first intersection times are studied for planar finite horizon Lorentz processes with a periodic configuration of scatterers. Their asymptotic behavior is analogous to the asymptotic behavior of the same quantities for the 2-d simple symmetric random walk (cf. classical results of Darling-Kac, 1957 and of Erdős-Taylor, 1960). Moreover, asymptotical distributions for phases in first hittings and in first intersections of Lorentz processes are also described. The results are also extended to the quasi-one-dimensional model of the linear Lorentz process. Subject classification: 37D50 billiards, 60F05 weak theorems

Random walks with internal degrees of freedom

TheK-property is demonstrated for a class of planar billiards satisfying Wojtkowskis principles. Their boundary may consist of convex-scattering, concave and linear pieces. Earlier Wojtkowski showed that these billards had non-zero Lyapunov exponents.

Recurrence properties of planar Lorentz process

Toric billiards with cylindric scatterers (briefly cylindric billards) generalize the class of Hamiltonian systems of elastic hard balls. In this paper a class of cylindric billiards is considered where the cylinders are “orthogonal” or more exactly: the constituent space of any cylindric scatterer is spanned by some of the (of course, orthogonal) coordinate vectors adapted to the euclidean torus. It is shown that the natural necessary condition for the K-property of such billiards is also sufficient.

On the

A conjecture is formulated and discussed which provides a necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup