Almost Countable Spectrum and Logarithmic Sarnak Conjecture

Abstract

In this paper, we introduce topological dynamical systems with almost countable spectrum. We prove that the Logarithmic Sarnak Conjecture holds for zero-entropy topological dynamical systems whose spectrum is almost countable. This class includes Anzai skew product on ${\mathbb{T}}^2$ over a rotation of ${\mathbb{T}}^1$, time-one maps of continuous suspension flows over rotations, systems with finite maximal pattern entropy, and bounded tame systems.