The \(K\)-property of \(N\) billiard balls. I
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.
discontinuous hyperbolic dynamical system
scientific article; zbMATH DE number 126364