Li-Yorke chaotic property of cookie-cutter systems

Alqahtani Bushra Abdulshakoor M; Weibin Liu; Alqahtani Bushra Abdulshakoor M; Weibin Liu

doi:10.3934/math.2022727

AIMS Mathematics

2022, Volume 7, Issue 7: 13192-13207. doi: 10.3934/math.2022727

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Research article

Li-Yorke chaotic property of cookie-cutter systems

Alqahtani Bushra Abdulshakoor M ¹,
Weibin Liu ^{2
,
,}

1.
Department of Mathematics, South China University of Technology, Guangzhou, 510640, China
2.
Department of Basic Courses, Guangzhou Maritime University, Guangzhou, 510725, China

Received: 25 January 2022 Revised: 07 April 2022 Accepted: 11 April 2022 Published: 10 May 2022
MSC : 11K55, 34C28

In this paper, we investigate mean Li-Yorke chaos along some sequence and Li-Yorke chaos for cookie-cutter systems. By applying bounded distortion and a locally $ \alpha $-H$ \ddot{\rm{o}} $lder condition, we show that the cookie-cutter set contains a mean Li-Yorke scrambled set along some sequence in which the Hausdorff dimension equals the Hausdorff dimension of the cookie-cutter set. That is to say, a cookie-cutter system is mean Li-Yorke chaotic along some sequence. Meanwhile, we proved that every mean Li-Yorke scrambled set is also a scrambled set; hence a cookie-cutter system is also Li-Yorke chaotic.
- cookie-cutter sets,
- mean Li-Yorke chaotic set,
- Hausdorff dimension,
- bounded distortion,
- Li-Yorke chaos
Citation: Alqahtani Bushra Abdulshakoor M, Weibin Liu. Li-Yorke chaotic property of cookie-cutter systems[J]. AIMS Mathematics, 2022, 7(7): 13192-13207. doi: 10.3934/math.2022727

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Abstract

In this paper, we investigate mean Li-Yorke chaos along some sequence and Li-Yorke chaos for cookie-cutter systems. By applying bounded distortion and a locally $ \alpha $-H$ \ddot{\rm{o}} $lder condition, we show that the cookie-cutter set contains a mean Li-Yorke scrambled set along some sequence in which the Hausdorff dimension equals the Hausdorff dimension of the cookie-cutter set. That is to say, a cookie-cutter system is mean Li-Yorke chaotic along some sequence. Meanwhile, we proved that every mean Li-Yorke scrambled set is also a scrambled set; hence a cookie-cutter system is also Li-Yorke chaotic.

References

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