Xiaoliang ZHOU, Risong LI

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 y: R+ \times X \rightarrow X and f: R+ \times Y \rightarrow Y) on the metric spaces X and Y, f \times g (resp. y \times f) is ergodically sensitive if and only if f or g (resp. y or f) is ergodically sensitive. Our results improve and extend some existing ones.

Anahtar Kelimeler:

Chaos in the sense of Devaney, topological transitivity, sensitivity

A note on chaos in product maps

Keywords:

Chaos in the sense of Devaney, topological transitivity, sensitivity,

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