The zero entropy locus for the Lozi maps

TL;DR

This paper investigates the parameter space of Lozi maps, identifying a specific region where the maps exhibit zero topological entropy and analyzing the dynamics related to period-two orbits and homoclinic points.

Contribution

The paper defines a region in the parameter space of Lozi maps where the topological entropy is zero and characterizes the dynamical behavior within this region.

Findings

Identifies a region in parameter space with zero topological entropy.

Shows that in a larger region, Lozi maps have a unique attracting period-two orbit.

Demonstrates the presence of homoclinic points in certain parameters.

Abstract

We study the zero entropy locus for the Lozi maps. We first define a region $R$ in the parameter space and prove that for the parameters in $R$, the Lozi maps have the topological entropy zero. $R$ is contained in a larger region where every Lozi map has a unique period-two orbit, and that orbit is attracting. It is easy to see that the zero entropy locus cannot coincide with that larger region since it contains parameters for which the fixed point of the corresponding Lozi map has homoclinic points.