Accumulation sets and zero entropy dynamics in the Lozi map

Abstract

For the Lozi map $L_{a,b}$, we consider parameter pairs for which the fixed 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 fixed 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 Štimac to a broader set of parameters.