A note on chaos in product maps

Authors : Risong LI, Xiaoliang ZHOU

Pages : 665-675

Doi:10.3906/mat-1101-71

View : 15 | Download : 5

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we mainly discuss how chaos conditions on semi-flows carry over to their products. We show that if two semi-flows insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(not-necessarily continuous); maps f: X \rightarrow X and g: Y \rightarrow Y insert ignore into journalissuearticles values(resp. semi-flows y: R+ \times X \rightarrow X and f: R+ \times Y \rightarrow Y); on the metric spaces X and Y, f \times g insert ignore into journalissuearticles values(resp. y \times f); is ergodically sensitive if and only if f or g insert ignore into journalissuearticles values(resp. y or f); is ergodically sensitive. Our results improve and extend some existing ones.
Keywords : Chaos in the sense of Devaney, topological transitivity, sensitivity