The set-valued mapping induced by a non-minimal transitive system is Li–Yorke chaotic

Let X be a metric space, (X, f) a discrete dynamical system, where f: X → X   is a continuous function. Let f¯ denote the natural extension of f to the space of all non-empty compact subsets of X endowed with a Hausdorff metric. In this paper, we prove that if f   is transitive and non-minimal, then f¯ is Li–Yorke’s chaos. Furthermore, if f is non-minimal M  -system, then f¯ has a s-scrambled set.