Chaos in product maps

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]