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.
Categories: math.DS