A note on chaos in product maps

In this paper, we mainly discuss how chaos conditions on semi-flows carry over to their products. We show that if two semi-flows (or even one of them) are sensitive, so does their product. On the other side, the product of two topologically transitive semi-flows need not be topologically transitive. We then provide several sufficient conditions under which the product of two chaotic semi-flows is chaotic in the sense of Devaney. Also, stronger forms of sensitivity and transitivity for product systems are studied. In particular, we introduce the notion of ergodic sensitivity and prove that for any given two (not-necessarily continuous) maps f: X rightarrow X and g: Y rightarrow Y (resp. semi-flows psi: R^+ times X rightarrow X and phi: R^+ times Y rightarrow Y) on the metric spaces X and Y, f times g (resp. psi times phi) is ergodically sensitive if and only if f or g (resp. psi or phi) is ergodically sensitive. Our results improve and extend some existing ones.