Almost Countable Spectrum and Logarithmic Sarnak Conjecture

2025/11/06 by Wen Huang, Huang, Wen, Min Tan +3
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2511.04419

openalex publication_date 2025/11/06 · openalex created_date 2025/11/08 · openalex updated_date 2026/07/28

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 \mathbbT² over a rotation of \mathbbT¹, time-one maps of continuous suspension flows over rotations, systems with finite maximal pattern entropy, and bounded tame systems.

Citations

• Independence, sequence entropy and mean sensitivity for invariant measures
• Möbius Disjointness for product flows of rigid dynamical systems and affine linear flows
• Polynomial mean complexity and Logarithmic Sarnak conjecture
• Sarnak's Conjecture for Sequences of Almost Quadratic Word Growth
• The logarithmic Sarnak conjecture for ergodic weights
• Measure complexity and Möbius disjointness
• Möbius disjointness for topological models of ergodic systems with discrete spectrum
• Equivalence of the logarithmically averaged Chowla and Sarnak conjectures
• The logarithmically averaged Chowla and Elliott conjectures for two-point correlations
• Infinite-step nilsystems, independence and complexity
• The quantitative behaviour of polynomial orbits on nilmanifolds
• Independence in topological and C*-dynamics

Related

• The logarithmic Sarnak conjecture for ergodic weights
• Polynomial mean complexity and Logarithmic Sarnak conjecture
• Oscillating sequences, Gowers norms and Sarnak's conjecture
• Fully oscillating sequences and weighted multiple ergodic limit
• Sarnak's conjecture implies the Chowla conjecture along a subsequence