The \(K\)-property of \(N\) billiard balls. I
type: Item
instance of: scholarly article
title: The \(K\)-property of \(N\) billiard balls. I
zbMATH Open document ID: 0759.58029
DOI: 10.1007/BF02100616
published in: Inventiones Mathematicae
publication date: 1993-03-03T00:00:00Z
full work available at URL: https://eudml.org/doc/144004
review text: This paper and its continuation ``The \(K\)-property of \(N\) billiard balls. II'' contain the proof of the Kolmogorov mixing property for the billiard system of \(N\) hard balls on torus, if only the dimension of the underlying torus is not less than \(N\). The proof is an induction on the number of balls \(N\). Actually, the present one of the two papers contains two thirds of the inductive step: Showing that, apart from a very tiny set from the viewpoint of measure theory and topology, the symbolic collision structures of the trajectories are combinatorially rich and checking the validity of the ``Chernov-Sinai Ansatz''. The missing third component of the induction --- the ``geometric-algebraic considerations'' as we call it --- will be presented in the second paper. It is just the part of the proof that requires the uncomfortable condition for the dimension of the torus. Important technical skills, two strengthened ``ball avoiding theorems'' --- which can also be considered as byproducts --- have been developed in Section 4.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37D99
zbMATH DE Number: 126364
zbMATH Keywords: Kolmogorov-mixing flow
zbMATH Keywords: discontinuous hyperbolic dynamical system
zbMATH Keywords: hard ball system
zbMATH Keywords: local stable invariant manifold
zbMATH Keywords: local unstable invariant manifolds
author: Nándor Simányi
MaRDI profile type: Publication
cites work: Q4150804
cites work: On the number of collisions in a hard sphere particle system in all space
cites work: Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems
cites work: Ergodic properties of semi-dispersing billiards. I. Two cylindric scatterers in the 3D torus
cites work: A ``transversal'' fundamental theorem for semi-dispersing billiards
cites work: The K-property of three billiard balls
cites work: Perturbed billiard systems II. Bernoulli properties
cites work: Dynamical systems with countably-multiple Lebesgue spectrum. II
cites work: Principles for the design of billiards with nonvanishing Lyapunov exponents
description: scientific article; zbMATH DE number 126364