Mean Li--Yorke chaos and multifractal analysis on subshifts

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.