Ergodicity in the linked twist map with oppositely oriented shears

TL;DR

This paper advances the understanding of ergodicity in linked twist maps by lowering shear parameter thresholds and demonstrating ergodicity with minimal twists, broadening the class of maps with proven ergodic behavior.

Contribution

It reduces the optimal shear parameter for ergodicity in linked twist maps and shows ergodicity with only one-fold twists, extending previous results.

Findings

Ergodicity established for shear parameter as low as 3.47.

Almost hyperbolicity proven for shear parameters greater than 2.

Improved bounds on shear parameters for ergodicity in linked twist maps.

Abstract

We reduce the earlier known optimal shear parameter for which ergodicity is established in the linked twist map with two linear shears in opposite sense, in the most general setting. Further, here we obtain ergodicity with possibly only one-fold twists in either lobe, while earlier results only applied for twist parameters at least 2. Almost hyperbolicity is easily established for shear parameters greater than 2, while in the most general setting of the linked twist map with both shears of equal magnitude, ergodicity was earlier established in the most general setting by Przytycki for shear parameters greater than 4.15 with at least two-fold twists in each lobe. Here we reduce this optimal shear parameter to 3.47 in the general setting. These techniques can be effected to make further improvements when additional assumptions are made on the dimensions of the strips or when the linked twist map is modified in other natural ways.