Invariant scrambled sets, uniform rigidity and weak mixing

TL;DR

This paper characterizes when transitive dynamical systems have dense invariant scrambled sets, linking their existence to fixed points and uniform rigidity, and introduces methods to construct such systems with specific properties.

Contribution

It provides a complete characterization of invariant scrambled sets in transitive systems and introduces new construction methods for weakly mixing, proximal, and uniformly rigid systems.

Findings

Dense invariant strongly scrambled sets exist iff the system has a fixed point.

Dense invariant δ-scrambled sets exist iff the system has a fixed point and is not uniformly rigid.

Methods are provided to construct weakly mixing, proximal, and uniformly rigid systems.

Abstract

We show that for a non-trivial transitive dynamical system, it has a dense Mycielski invariant strongly scrambled set if and only if it has a fixed point, and it has a dense Mycielski invariant $\delta$-scrambled set for some $\delta>0$ if and only if it has a fixed point and not uniformly rigid. We also provide two methods for the construction of completely scrambled systems which are weakly mixing, proximal and uniformly rigid.