A Bernoulli linked-twist map in the plane

TL;DR

This paper proves that a specific Lebesgue measure-preserving linked-twist map in the plane is metrically isomorphic to a Bernoulli shift, establishing strong mixing properties for the first time for such a map on a manifold beyond the two-torus.

Contribution

It demonstrates the Bernoulli property for an explicitly defined linked-twist map on a plane, extending ergodic theory results to new manifold settings.

Findings

The linked-twist map is metrically isomorphic to a Bernoulli shift.

The map exhibits strong mixing properties.

First such result for a linked-twist map outside the two-torus.

Abstract

We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a manifold other than the two-torus. Our work builds on that of Wojtkowski who established an ergodic partition for this example using an invariant cone-field in the tangent space.