Stable sets and mean Li–Yorke chaos in positive entropy systems

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically “rather big” set such that a multivariant version of mean Li–Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li–Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.