Almost Countable Spectrum and Logarithmic Sarnak Conjecture

TL;DR

This paper introduces a class of topological dynamical systems with almost countable spectrum and proves the Logarithmic Sarnak Conjecture for zero-entropy systems within this class, including several well-known examples.

Contribution

It establishes the validity of the Logarithmic Sarnak Conjecture for a broad class of zero-entropy systems with almost countable spectrum, expanding previous results.

Findings

Logarithmic Sarnak Conjecture holds for systems with almost countable spectrum.

Includes systems like Anzai skew products, suspension flows, and tame systems.

Validates conjecture for a wide class of zero-entropy systems.

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.