Combinatorial embedding of chain transitive zero-dimensional systems into chaos

TL;DR

This paper demonstrates that zero-dimensional chain transitive systems can be embedded into densely chaotic systems, constructing an invariant Mycielski set with rich dynamical properties, including transitivity, proximality, and recurrence.

Contribution

It introduces a method to embed zero-dimensional chain transitive systems into densely chaotic systems with a specifically constructed invariant Mycielski set.

Findings

Constructed a dense, invariant Mycielski set with transitivity and recurrence.

Embedded zero-dimensional chain transitive systems into densely chaotic systems.

Established bidirectional proximality and recurrence properties of the set.

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.