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

Anna Cima , Armengol Gasull i Embid , Víctor Mañosa Fernández , Francesc Mañosas Capellades

• We consider the family of piecewise linear maps Fa,b(x, y) = (|x| − y + a, x − |y| + b), where (a, b) ∈ R2. 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 ∈ R there exists a compact graph , which is invariant under the map F, such that for each (x, y) ∈ R2 there exists n ∈ N (that may depend on x) such that Fn a,b(x, y) ∈ . We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all (a, b) ∈ R2 obtaining, among other results, a full characterization of when Fa,b| has positive or zero entropy