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Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Volume 244 / 2019

Livio Flaminio, Krzysztof Frączek, Joanna Kułaga-Przymus, Mariusz Lemańczyk Studia Mathematica 244 (2019), 43-97 MSC: 37A05, 37A17, 37A45, 37D40, 11N37. DOI: 10.4064/sm170512-25-9 Published online: 14 May 2018

Abstract

Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma \lt G $ a lattice. We prove that each ergodic diffeomorphism $ \phi(x\Gamma)=uA(x)\Gamma $ on the nilmanifold $ G/\Gamma $, where $ u\in G $ and $ A\colon G\to G $ is a unipotent automorphism satisfying $ A(\Gamma)=\Gamma $, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow:

(i) Sarnak’s conjecture on Möbius orthogonality holds in every uniquely ergodic model of each ergodic affine unipotent diffeomorphism;

(ii) for ergodic affine unipotent diffeomorphisms themselves, Möbius orthogonality holds on so-called typical short intervals: \[ \frac1M\sum_{M\leq m \lt 2M}\bigg|\frac1H\sum_{m\leq n \lt m+H} f(\phi^n(x\Gamma))\boldsymbol{\mu} (n)\bigg|\to 0 \] as $ H\to\infty $ and $ H/M\to0 $ for each $ x\Gamma\in G/\Gamma $ and each $ f\in C(G/\Gamma) $.

In particular, (i) and (ii) hold for ergodic niltranslations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function $\boldsymbol{\mu}$ on a typical short interval.

We also study the problem of lifting the AOP property to induced actions, and derive some applications to uniform distribution.

Authors

  • Livio FlaminioUnité Mixte de Recherche CNRS 8524 Unité de Formation et Recherche de Mathématiques
    Université de Lille
    F-59655 Villeneuve d’Ascq Cedex, France
    e-mail
  • Krzysztof FrączekFaculty of Mathematics and
    Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail
  • Joanna Kułaga-PrzymusFaculty of Mathematics and
    Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail
  • Mariusz LemańczykFaculty of Mathematics and
    Computer Science
    Nicolaus Copernicus University
    Chopina 12/18
    87-100 Toruń, Poland
    e-mail

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