Directional stable sets and mean Li-Yorke chaos in positive directional entropy systems

Abstract

It is shown that if a Z q -system has a measure with positive directional measure-theoretic entropy then it is multivariant directional mean Li-Yorke chaotic along the corresponding direction. Meanwhile, the notion of directional measuretheoretic entropy tuples is introduced and many properties of them are studied. It is also proved that for any ergodic invariant measure on the Z q -system, the intersection of the set of directional measure-theoretic entropy tuples with the set of directional asymptotic tuples is dense in the set of directional measure-theoretic entropy tuples.