Some Remarks on the Notion of Bohr Chaos and Invariant Measures

TL;DR

This paper explores the concept of Bohr chaos in dynamical systems, establishing new results about its presence in systems with the specification property and identifying obstructions related to invariant measures.

Contribution

It answers an open question about Bohr chaos in non-uniquely ergodic systems and proves that systems with the specification property are Bohr chaotic, providing new insights and proofs.

Findings

Systems with the specification property are Bohr chaotic.

Non-uniquely ergodic systems of positive entropy can be Bohr chaotic.

Systems with fewer than a continuum of ergodic measures cannot be Bohr chaotic.

Abstract

The notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic.