Related papers: The Mobius disjointness conjecture for distal flow…

We formulate and prove a finite version of Vinogradov's bilinear sum inequality. We use it together with Ratner's joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on $\Gamma \backslash…

We prove that the M\"{o}bius function is linearly disjoint from an analytic skew product on the $2$-torus. These flows are distal and can be irregular in the sense that their ergodic averages need not exist for all points. The previous…

Let $\mathbb{T}$ be the unit circle and $\Gamma\setminus G$ the 3-dimensional Heisenberg nilmanifold. We prove that the M\"obius function is linearly disjoint from a class of distal skew products on $\mathbb{T}\times\Gamma\setminus G$.…

Furstenberg's flow on the infinite-dimensional torus $\mathbb{T}^\omega$ is defined by \[ T (x_1, x_2, \ldots, x_\nu, \ldots) = (x_1 + \alpha, x_2 + h(x_1), \ldots, x_\nu + h(x_1 + (\nu-2)\beta), \ldots) \] with $\alpha\in \mathbb{R}$…

As the main theorem, it is proved that a collection of minimal $PI$-flows with a common phase group and satisfying a certain algebraic condition is multiply disjoint if and only if the collection of the associated maximal equicontinuous…

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic…

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

We study the distribution of the sequence of elements of the discrete dynamical system generated by the M\"obius transformation $x \mapsto (ax + b)/(cx + d)$ over a finite field of $p$ elements. Motivated by a recent conjecture of P.…

We say that two arithmetic functions f and g form a Mobius pair if f(n) = \sum_{d \mid n} g(d) for all natural numbers n. In that case, g can be expressed in terms of f by the familiar Mobius inversion formula of elementary number theory.…

We show that ergodic dynamical systems generated by infinitely divisible stationary processes are disjoint in the sense of Furstenberg with distally simple systems and systems whose maximal spectral type is singular with respect to the…

We prove that on the typical translation surface the flow in almost every pair of directions are not isomorphic to each other and are in fact disjoint. It was not known if there were any translation surfaces other than torus covers with…

In this paper we will give a new proof of the monotonicity of Wasserstein distances of two diffusions under super Ricci flow. Our proof is based on the coupling method of B.Andrew and J.Clutterbuck. The same method can also be applied to…

We prove the existence of periodic orbits for steady $C^\omega$ Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We…

Biological functionality of cellular aggregates is largely influenced by the activity and displacements of individual constituent cells. From a theoretical perspective this activity can be characterized by hydrodynamic transport…

We prove a function field version of Chowla's conjecture on the autocorrelation of the Mobius function in the limit of a large finite field.

Let $\mathcal F$ be a co-oriented $C^2$ foliation on a closed, oriented 3-manifold. We show that $T\mathcal F$ can be perturbed to a contact structure with Reeb flow transverse to $\mathcal F$ if and only if $\mathcal F$ does not support an…

We explore the `Moebius disjointness property' in the special context of rank-one transformations and verify this phenomenon for many of the `classical' models

The Ratner property, a quantitative form of divergence of nearby trajectories, is a central feature in the study of parabolic homogeneous flows. Discovered by Marina Ratner and used in her 1980th seminal works on horocycle flows, it pushed…

This paper is concerned with Loxodromic Mobius Transformations with disjoint Axes in a Kleinian Group. We study mainly the distance between their axes, and give some estimates about their translation lengths.

We analyze the fluctuations of the dissipated energy in a simple and general model where dissipation, diffusion and driving are the key ingredients. The large deviation function for the dissipation follows from hydrodynamic fluctuation…