Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two

TL;DR

This paper studies the dynamics of a class of high-dimensional billiard systems with random scatterers, proving that they are hyperbolic, recurrent, ergodic, and exhibit complex chaotic behavior under broad conditions.

Contribution

It establishes the hyperbolicity and ergodic properties of quenched random Lorentz tubes in dimensions higher than two, extending understanding of their chaotic dynamics.

Findings

All systems in the ensemble are hyperbolic.

Almost every system is recurrent and ergodic.

Systems exhibit higher chaotic properties.

Abstract

We consider the billiard dynamics in a non-compact set of R^d that is constructed as a bi-infinite chain of translated copies of the same d-dimensional polytope. A random configuration of semi-dispersing scatterers is placed in each copy. The ensemble of dynamical systems thus defined, one for each global realization of the scatterers, is called `quenched random Lorentz tube'. Under some fairly general conditions, we prove that every system in the ensemble is hyperbolic and almost every system is recurrent, ergodic, and enjoys some higher chaotic properties.