Li-Yorke chaotic property of cookie-cutter systems

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 α-Ho¨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.

Keywords

• cookie-cutter sets
• mean li-yorke chaotic set
• hausdorff dimension
• bounded distortion
• li-yorke chaos