Loss of Exponential Mixing in a Non-Monotonic Toral Map

TL;DR

This paper studies a specific 2-torus map that preserves measure and exhibits strong mixing with polynomial decay of correlations, highlighting how non-monotonicity affects exponential mixing.

Contribution

It demonstrates strong mixing and polynomial decay in a non-monotonic toral map, linking chaotic billiards results to this class of piecewise linear systems.

Findings

Establishes polynomial decay of correlations for the map.

Identifies the boundary of ergodicity with null measure sets.

Shows how perturbations can create elliptic islands.

Abstract

We consider a Lebesgue measure preserving map of the 2-torus, given by the composition of orthogonal tent shaped shears. We establish strong mixing properties with respect to the invariant measure and polynomial decay of correlations for Holder observables, making use of results from the chaotic billiards literature. The system serves as a prototype example of piecewise linear maps which sit on the boundary of ergodicity, possessing null measure sets around which mixing is slowed and which birth elliptic islands under certain perturbations.