tailieunhanh - Chaos in product maps

We discuss how chaos conditions on maps carry over to their products. First we give a counterexample showing that the product of two chaotic maps (in the sense of Devaney) need not be chaotic. We then remark that if two maps (or even one of them) exhibit sensitive dependence on initial conditions, so does their product, likewise, if two maps possess dense periodic points, so does their product. | Turk J Math 34 (2010) , 593 – 600. ¨ ITAK ˙ c TUB doi: Chaos in product maps Nedim De˘girmenci, S¸ahin Ko¸cak Abstract We discuss how chaos conditions on maps carry over to their products. First we give a counterexample showing that the product of two chaotic maps (in the sense of Devaney) need not be chaotic. We then remark that if two maps (or even one of them) exhibit sensitive dependence on initial conditions, so does their product; likewise, if two maps possess dense periodic points, so does their product. On the other side, the product of two topologically transitive maps need not be topologically transitive. We then give sufficient conditions under which the product of two chaotic maps is chaotic in the sense of Devaney [6]. Key Words: Devaney’s chaos, topological transitivity, sensitive dependence on initial conditions. 1. Introduction Let X and Y be two metric spaces and f : X → X, g : Y → Y two maps, which we assume not to be continuous in general, but chaotic in the sense of Devaney (which we explain instantly). It is natural to ask whether their product f × g : X × Y → X × Y is also chaotic (in the same sense). We show by counter-example that the answer is in the negative. We then discuss the transfer of the sub-conditions of chaos and finally give some simple sufficient conditions making the product chaotic. These conditions are satisfied for many known chaotic maps. Now we first recall the chaos conditions for a not-necessarily continuous map f : X → X, X being a metric space with metric d . The discrete dynamical system (X, f) and the map f are used as synonyms in this work, so that phrases such as “The map f is chaotic” or “The discrete dynamical system (X, f) exhibits chaos” are used in the same sense. Definition 1 Sensitive dependence on initial conditions: A (not-necessarily continuous) map f : X → X is called sensitively dependent on initial conditions, if there exists ε > 0 such that, for any x ∈ X , and for any neighborhood U