Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
Tom 244 / 2019
Streszczenie
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.
