Accumulation sets and zero entropy dynamics in the Lozi map.

Abstract

For the Lozi map La,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 ℓ be the set of accumulation points of the unstable manifold WXu that do not lie on WXu. We construct a polygon D whose forward images under La,b form nested sequences of sets that eventually become trapping. We show that this geometric construction gives a characterization of ℓ as the intersection of these iterates. Using this structure, we prove that the non-wandering set for La,b2 is contained in the union of ℓ and the set of fixed points of La,b. As a consequence, the Lozi map, restricted to the complement of ℓ in the plane, has zero topological entropy. This result extends a recent one of Misiurewicz and Štimac to a broader set of parameters.