Infinite-horizon Lorentz tubes and gases: recurrence and ergodic properties

TL;DR

This paper constructs and analyzes classes of two-dimensional Lorentz systems with infinite horizon, demonstrating their chaotic, recurrent, hyperbolic, and ergodic properties, and shows that such systems are prevalent within certain families.

Contribution

It introduces new classes of aperiodic Lorentz systems with infinite horizon that are proven to be chaotic, ergodic, and hyperbolic, expanding understanding of their dynamical properties.

Findings

Lorentz systems with infinite horizon are recurrent and ergodic.

The first-return map to any scatterer is K-mixing.

Chaotic Lorentz gases are dense and uncountable within certain families.

Abstract

We construct classes of two-dimensional aperiodic Lorentz systems that have infinite horizon and are 'chaotic', in the sense that they are (Poincar\'e) recurrent, uniformly hyperbolic, ergodic, and the first-return map to any scatterer is $K$-mixing. In the case of the Lorentz tubes (i.e., Lorentz gases in a strip), we define general measured families of systems (\emph{ensembles}) for which the above properties occur with probability 1. In the case of the Lorentz gases in the plane, we define families, endowed with a natural metric, within which the set of all chaotic dynamical systems is uncountable and dense.