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Dergiler / Turkish Journal of Mathematics / 2013 / Cilt: 37 - Sayı: 4

A note on chaos in product maps

Xiaoliang ZHOU
Risong LI
Sayfa
665–675
Yayın Tarihi
2022-11-05
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Our results improve and extend some existing ones.

📌 3 Maddede Bu Makale

🎯
Amaç & Hipotez:In this paper, we mainly discuss how chaos conditions on semi-flows carry over to their products.
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Temel Bulgular & Veri:We show that if two semi-flows (or even one of them) are sensitive, so does their product.
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Sonuç & Katkı:Our results improve and extend some existing ones.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:Chaos in the sense of Devaney · topological transitivity · sensitivity

Özet

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.

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 (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.

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Bu Makaleye Atıf Yap

ZHOU, X., LI, R. (2022). A note on chaos in product maps. Turkish Journal of Mathematics, Cilt: 37 - Sayı: 4, 665–675. https://ajindex.com/articles/608368

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Kaynakça (11)

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