Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

TL;DR

This paper studies the dynamics of a family of piecewise linear planar maps, revealing periodic behaviors for non-negative parameters and invariant graphs with complex dynamics for negative parameters, including entropy characterization.

Contribution

It provides a complete description of invariant graphs and dynamics for all parameters, extending understanding of generalized Lozi-type maps.

Findings

For a ≥ 0, all orbits are eventually periodic with at most three behaviors.

For a < 0, invariant graphs exist for all b, capturing the dynamics of the map.

The paper characterizes when the restricted map has positive or zero entropy.

Abstract

We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $\Gamma,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in \Gamma.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_{\Gamma}$ has positive or zero entropy.