Extensions of exact and K-mixing dynamical systems

TL;DR

This paper studies extensions of non-singular dynamical systems that are exact or K-mixing, providing decomposition theorems and applying results to show many Lorentz gases are K-mixing.

Contribution

It introduces decomposition theorems for fiber-surjective and fiber-bijective extensions of dynamical systems, with applications to Lorentz gases.

Findings

Proved Exact and K-mixing Decomposition Theorems.

Applied results to show many Lorentz gases are K-mixing.

Extended understanding of skew product systems in dynamical systems.

Abstract

We consider extensions of non-singular maps which are exact, respectively K-mixing, or at least have a decomposition into positive-measure exact, respectively K-mixing, components. The fibers of the extension spaces have countable (finite or infinite) cardinality and the action on them is assumed surjective or bijective. We call these systems, respectively, fiber-surjective and fiber-bijective extensions. Technically, they are skew products, though the point of view we take here is not the one generally associated with skew products. Our main results are an Exact and a K-mixing Decomposition Theorem. The latter can be used to show that a large number of periodic Lorentz gases (the term denoting here general group extensions of Sinai billiards, including Lorentz tubes and slabs, in any dimension) are K-mixing.