Accumulation sets and zero entropy dynamics in the Lozi map

TL;DR

This paper characterizes the accumulation set of the unstable manifold in Lozi maps and shows that, outside this set, the dynamics have zero topological entropy, extending previous results to more parameters.

Contribution

It introduces a geometric construction of the accumulation set and proves zero entropy for the map outside this set for a broader parameter range.

Findings

The accumulation set    is characterized as the intersection of nested images of a polygon.

The non-wandering set of the Lozi map squared is contained in the union of    and fixed points.

The Lozi map has zero topological entropy outside the accumulation set for a wider parameter set.

Abstract

For the family of Lozi maps $L_{a,b}$, we consider parameter pairs for which the f\mbox{}ixed point $X$ has no homoclinic points and the period-two orbit $\{P,P'\}$ is attracting. For such parameters, let $\ell$ be the set of accumulation points of the unstable manifold $W_X^u$ that do not lie on $W_X^u$. We construct a polygon $\mathcal{D}$ whose forward images under $L_{a,b}$ form nested sequences of sets that eventually become trapping. We show that this geometric construction gives a characterization of $\ell$ as the intersection of these iterates. Using this structure, we prove that the non-wandering set for $L_{a,b}^2$ is contained in the union of $\ell$ and the set of f\mbox{}ixed points of $L_{a,b}$. As a consequence, the Lozi map, restricted to the complement of $\ell$ in the plane, has zero topological entropy. This result extends a recent one of Misiurewicz and \v{S}timac to a broader set of parameters.