Markov partitions for two-dimensional hyperbolic billiards. Statistical properties of two-dimensional hyperbolic billiards. Statistical properties of Lorentz gas with periodic configuration of scatterers.
Markov partitions for dispersed billiards. The fundamental theorem of the theory of scattering billiards. A central limit theorem for scattering billiards. Variational principle for periodic trajectories of hyperbolic billiards. Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon. Quantizing a classically ergodic system: Sinai’s billiard and the KKR method. Lectures on spaces of nonpositive curvature. Lecture given at the meeting in the Fields institute dedicated to his 60th birthday. The isoperimetric problem and estimates of a length of a curve on a surface. A theorem on triangles in a metric space and some applications of it. This process is experimental and the keywords may be updated as the learning algorithm improves.Ī.D. These keywords were added by machine and not by the authors. In particular, this approach helped to solve an old problem of whether the number of collisions in a hard ball model is bounded from above by a quantity depending only on the system (and thus uniform for all initial conditions). Nevertheless, this method allows to transforms a certain type of problems about billiards into purely geometric statements and a problem looking difficult in its billiard clothing sometimes turns into a relatively easy statement (by the modern standards of metric geometry). These spaces are not even topological manifolds: they are lengths spaces of curvature bounded above in the sense of A.
The approach is based on representing billiard trajectories as geodesics in appropriate spaces. Arnold’s old idea that hard ball models of statistical physics can be “considered as the limit case of geodesic flows on negatively curved manifolds (the curvature being concentrated on the collisions hypersurface)”. This section contains a survey of a few results obtained by a particular realization of V.
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