Bohr chaoticity of topological dynamical systems

TL;DR

This paper introduces Bohr chaoticity as a new topological invariant for dynamical systems, demonstrating its presence in systems with horseshoes and positive entropy, while showing uniquely ergodic systems lack this property.

Contribution

It defines Bohr chaoticity, explores its properties, and establishes its presence or absence in various classes of topological dynamical systems.

Findings

Bohr chaoticity is a topological invariant opposite to Sarnak's conjecture requirements.

Systems with horseshoes and positive entropy are Bohr chaotic.

Uniquely ergodic systems are not Bohr chaotic.

Abstract

We introduce the notion of Bohr chaoticity, which is a topological invariant for topological dynamical systems, and which is opposite to the property required by Sarnak's conjecture. We prove the Bohr chaoticity for all systems which have a horseshoe and for all toral affine dynamical systems of positive entropy, some of which don't have a horseshoe. But uniquely ergodic dynamical systems are not Bohr chaotic.