Loss of Exponential Mixing in a Non-Monotonic Toral Map

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.