Some remarks on the notion of Bohr chaos and invariant measures
Volume 271 / 2023
Studia Mathematica 271 (2023), 347-359
MSC: Primary 37A05; Secondary 37A46, 37B02, 37B05, 37B10.
DOI: 10.4064/sm230103-13-5
Published online: 26 July 2023
Abstract
The notion of Bohr chaos was introduced by Fan et al. (2021, 2022). We answer a question raised by Fan et al. (2022) 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 in this line of thought give an independent proof (and strengthening) of Theorem 1 of Fan et al. (2022). In addition, we present an obstruction to Bohr chaos: a system with fewer than continuum many ergodic invariant probability measures cannot be Bohr chaotic.
