Strange attractors for the family of orientation preserving Lozi maps

Abstract

We extend the result of Michal Misiurewicz assuring the existence of strange attractors for the parametrized family $\{f_{(a,b)}\}$ of orientation reversing Lozi maps to the orientation preserving case. That is, we rigorously determine an open subset of the parameter space for which an attractor ${\mathcal{A}}_{(a,b)}$ of $f_{(a,b)}$ always exists and exhibits chaotic properties. Moreover, we prove that the attractor is maximal in some open parameter region, and arises as the closure of the unstable manifold of a fixed point, on which $f_{(a,b)}{|}_{{\mathcal{A}}_{(a,b)}}$ is mixing. We also show that ${\mathcal{A}}_{(a,b)}$ vary continuously with parameter $(a,b)$ in the Hausdorff metric.