Chaos in product maps
We discuss how chaos conditions on maps carry over to their products. First we give a counterexample showing that the pro\-duct 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].
Anahtar Kelimeler:
Devaney's chaos, topological transitivity, sensitive dependence on initial conditions
Chaos in product maps
We discuss how chaos conditions on maps carry over to their products. First we give a counterexample showing that the pro\-duct 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].
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- a non-negative k such that fk(x)∈ V, that is, if every pair of non-empty open subsets of X shares a periodic orbit. (If f is continuous and X non-Şnite, then this property implies chaos in the sense of Devaney by [2]; see also [10]).
- Theorem 3 Let X be any metric space and assume that the (not-necessarily continuous) map f : X→ X has the Touhey property. Let g : Y→ Y be a not-necessarily continuous, chaotic and topologically mixing map on the metric space Y. Then f× g : X × Y → X × Y is chaotic. Proof.
- property implies denseness of periodic points of f, hence, as the periodic points of g are also dense, we have denseness of periodic points of f× g. The transitivity of f × g can be seen as in the preceding proof.
- has no periodic points, but it is topologically transitive and sensitively dependent on initial conditions.
- ISSN: 1300-0098
- Yayın Aralığı: Yılda 6 Sayı
- Yayıncı: TÜBİTAK
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