Combinatorial embedding of chain transitive zero-dimensional systems into chaos

Abstract

We show that a zero-dimensional chain transitive dynamical system can be embedded into a densely uniformly chaotic system, with dense uniformly chaotic set $K$. We concretely construct a Mycielski set $K$ that is also invariant. Furthermore, every point in $K$ is positively and negatively transitive. The uniform proximality and recurrence of $K$ are also bidirectional.