Möbius disjointness for $C^{1 + \varepsilon}$ skew products

Möbius disjointness for analytic skew products

We show that the Möbius function is disjoint to every analytic skew product dynamical system on the two-torus over a rotation of the circle.

Möbius disjointness for skew products on a circle and a nilmanifold

Let $\mathbb{T}$ be the unit circle and $Γ\backslash G$ the $3$-dimensional Heisenberg nilmanifold. We prove that a class of skew products on $\mathbb{T} \times Γ\backslash G$ are distal, and that the Möbius function is linearly disjoint from these skew products. This verifies the Möbius Disjointness Conjecture of Sarnak.

Möbius Disjointness for Skew Products on the Heisenberg Nilmanifold

We prove that the Möbius function is disjoint to all Lipschitz continuous skew product dynamical systems on the 3-dimensional Heisenberg nilmanifold over a minimal rotation of the 2-dimensional torus.

Möbius disjointness for non-uniquely ergodic skew products

For $τ>2$, let $T$ be a $C^τ$ skew product map of the form $(x+α,y+h(x))$ on $\mathbb T^2$ over a rotation of the circle. We show that if $T$ preserves a measurable section, then it is disjoint to the Möbius sequence. This in particular implies that any non-uniquely ergodic $C^τ$ skew product map on $\mathbb T^2$ has a finite index factor that is disjoint to the Möbius sequence.

Measure complexity and Möbius disjointness

In this paper, the notion of measure complexity is introduced for a topological dynamical system and it is shown that Sarnak's Möbius disjointness conjecture holds for any system for which every invariant Borel probability measure has sub-polynomial measure complexity. Moreover, it is proved that the following classes of topological dynamical systems $(X,T)$ meet this condition and hence satisfy Sarnak's conjecture: (1) Each invariant Borel probability measure of $T$ has discrete spectrum. (2) $T$ is a homotopically trivial $C^\infty$ skew product system on $\mathbb{T}^2$ over an irrational ro…

动力学中的刚性性与Möbius不相交性

Rigidity in dynamics and Möbius disjointness

本文证明若动力系统的所有不变测度满足特定刚性条件，则Sarnak猜想成立，并推广至可数个遍历测度情形，覆盖区间交换映射、环面斜积与流等新案例，改进多项早期结果。