Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $  \beta  $-transformation

Yuanfen Xiao

doi:10.3934/dcds.2020267

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2021, Volume 41, Issue 2: 525-536. Doi: 10.3934/dcds.2020267

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Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation

Yuanfen Xiao

Department of Mathematics, University of Science and Technology of China, Hefei, 230026, Anhui, China

Received: November 30, 2019

Revised: March 31, 2020

Early access: July 2020

Published: February 2021

The author is supported by NNSF of China grant 11871228

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Abstract

We construct a mean Li-Yorke chaotic set along polynomial sequences (the degree of this polynomial is not less than three) with full Hausdorff dimension and full topological entropy for $ \beta $-transformation. An uncountable subset $ C $ is said to be a mean Li-Yorke chaotic set along sequence $ \{a_n\} $, if both

$ \begin{equation*} \liminf\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y )) = 0 \text{ and } \limsup\limits_{N\to \infty}\frac{1}{N}\sum\limits_{j = 1}^{N}d(f^{a_j}(x),f^{a_j}(y ))>0 \end{equation*} $

hold for any two distinct points $ x $ and $ y $ in $ C $.

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Mathematics Subject Classification: Primary:54H20, 37C45, 37D45;Secondary:37B40.

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