On the Invariance of Li-Yorke Chaos of Interval Maps

Abstract

In their celebrated "Period three implies chaos" paper, Li and Yorke proved that if a continuous interval map f has a period 3 point then there is an uncountable scrambled set S on which f has very complicated dynamics. One question arises naturally: Can this set S be chosen invariant under f? The answer is positive for turbulent maps and negative otherwise. In this note, we shall use symbolic dynamics to achieve our goal. In particular, we obtain that the tent map T(x) = 1 - |2x-1| on [0, 1] has a dense uncountable invariant 1-scrambled set which consists of transitive points.