A Continuous Family of Non-Monotonic Toral Mixing Maps

TL;DR

This paper proves the mixing property for a family of Lebesgue measure-preserving toral maps with non-monotonic shear components, modeling complex fluid mixing behaviors without relying on finite Markov partitions.

Contribution

It introduces a novel approach to establish mixing in a family of maps lacking finite Markov partitions, extending the understanding of non-monotonic toral mixing systems.

Findings

Established strong mixing properties for the family of maps.

Demonstrated the approach's applicability to systems without finite Markov partitions.

Identified challenges and potential for extending mixing results to broader classes.

Abstract

We establish the mixing property for a family of Lebesgue measure preserving toral maps composed of two piecewise linear shears, the first of which is non-monotonic. The maps serve as a basic model for the `stretching and folding' action in laminar fluid mixing, in particular flows where boundary conditions give rise to non-monotonic flow profiles. The family can be viewed as the parameter space between two well known systems, Arnold's Cat Map and a map due to Cerbelli and Giona, both of which possess finite Markov partitions and straightforward to prove mixing properties. However, no such finite Markov partitions appear to exist for the present family, so establishing mixing properties requires a different approach. In particular we follow a scheme of Katok and Strelcyn, proving strong mixing properties with respect to the Lebesgue measure on two open parameter spaces. Finally we comment on the challenges in extending these mixing windows and the potential for using the same approach to prove mixing properties in similar systems.