We consider the family of piecewise linear maps \(\begin{aligned} F_{a,b}(x,y)=\left( |x| - y + a, x - |y| + b\right) , \end{aligned}\) 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.
