Combinatorial embedding of chain transitive zero-dimensional systems into chaos

In this paper, we construct examples of transitive Li–Yorke chaotic zero-dimensional topological dynamical systems, which should be able to accommodate of a variety of subsystems. In fact, we show that a zero-dimensional chain transitive dynamical system can be embedded into a transitive Li–Yorke chaotic system with a dense scrambled set, and with the same topological entropy. Our work reflects the notion of uniform chaos, which was recently developed. In addition, our work reflects the recent work of invariance of scrambled sets in general topological setting. In these works, Mycielski sets play an essential role. Actually, we concretely construct a Mycielski set K such that it is densely uniformly chaotic, and also invariant in both directions. Furthermore, every point in K is positively and negatively transitive. The uniform proximality and recurrence of K are also bidirectional.