Mean Li--Yorke chaos and multifractal analysis on subshifts

TL;DR

This paper investigates the multifractal properties and entropy characteristics of historic sets in irreducible shifts of finite type, revealing that certain Li-Yorke pair sets have zero Bowen entropy but full packing entropy, extending beyond ergodic theory.

Contribution

It introduces a novel multifractal framework to analyze entropy of Li-Yorke pairs in shifts of finite type, providing new insights beyond traditional ergodic theory.

Findings

Bowen entropy of mean Li-Yorke pair sets is zero.

Packing entropy of these sets is full.

Results extend beyond classical ergodic theory.

Abstract

In the present paper, we use the generalized multifractal framework introduced by Olsen to study the Bowen entropy and packing entropy of historic sets with typical weights over aperiodic and irreducible shifts of finite type. Following those results and a transfer from almost everywhere to everywhere, we show that for each point $\omega$ in a irreducible shift of finite type $\Sigma_A$, the Bowen entropy of the set consisting of all the points that are mean Li-Yorke pairs with $\omega$ is $0$, and its packing entropy is full. This result is beyond the ergodic theory. Also, by the transfer from almost everywhere to everywhere, we show that for each point $\omega$ in a irreducible shift of finite type $\Sigma_A$, the Bowen entropy of the set consisting of all the points that are Li-Yorke pairs with $\omega$ is full. This result is also beyond the ergodic theory.