Invariant Graphs and Dynamics of a Family of Continuous Piecewise Linear Planar Maps

We consider the family of piecewise linear maps F-a,F-b(x,y)=(|x|-y+a,x-|y|+b), where (a,b)is an element of 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 >= 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 is an element of R there exists a compact graph Gamma, which is invariant under the map F, such that for each (x,y)is an element of R-2 there exists n is an element of N (that may depend on x) such that F-a,b(n)(x,y)is an element of 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)is an element of R-2 obtaining, among other results, a full characterization of when F-a,F-b|Gamma has positive or zero entropy.