Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds
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DOI：
10.4064/sm170512-25-9
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发表时间：
2016-09
影响因子：
0.8
通讯作者：
L. Flaminio;Krzysztof Frkaczek;J. Kułaga-Przymus;M. Lema'nczyk
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文献类型：
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作者：
L. Flaminio;Krzysztof Frkaczek;J. Kułaga-Przymus;M. Lema'nczyk
设G $是连通的单连通幂零李群，$ \Gamma < G $是格.本文证明了在零流形$ G/\Gamma $上的每个遍历自同构$ \phi（x\Gamma）=uA（x）\Gamma $，其中$ u\in G $和$ A：G\to G $是满足$ A（\Gamma）=\Gamma $的单幂自同构，具有渐近正交幂（AOP）性质.由此产生两个后果：（i）关于Mobius正交性的Sarnak猜想在遍历仿射幂单同态的每个唯一遍历模型中成立;（ii）对于遍历仿射幂单同态本身，Mobius正交性在所谓的典型短区间上成立：$\frac1 M\sum_{M\leq m<2 M}\left|\frac1H\sum_{m\leq n<m+H} f（\phi^n（x\Gamma））\mu（n）\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ x\Gamma\in G/\Gamma $ and each $ f\in C（G/\Gamma）$.特别地，（i）和（ii）中的结果对遍历零平移成立。此外，我们还证明了每个零序列在一个典型的短区间上与Mobius函数$\mu$正交.我们还研究了AOP性质对诱导作用的提升问题，并给出了在均匀分布上的一些应用。
Let $ G $ be a connected, simply connected nilpotent Lie group and $ \Gamma < 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: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 Mobius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Mobius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(\phi^n(x\Gamma))\mu (n)\right|\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, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Mobius function $\mu$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.