Rotation sets and almost periodic sequences

TL;DR

This paper investigates the rotational behavior of minimal sets in torus homeomorphisms, demonstrating that their rotation sets can be highly complex, including non-convex and plane-separating continua, unlike the restrictions on the entire torus.

Contribution

It introduces a method to realize diverse rotation sets on minimal sets using symbolic dynamics and irregular Toeplitz sequences, expanding understanding of rotational behaviors.

Findings

Rotation sets on minimal sets can be non-convex and plane-separating.

Restrictions on rotation sets do not apply to minimal sets.

Construction of rotational horseshoes enables realization of complex rotation sets.

Abstract

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.