Sarnak's Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow
Univ Rouen Normandie（鲁昂诺曼底大学）；CNRS, Department of Mathematics, LMRS UMR 6085 F-76000, Rouen, France（法国国家科学研究中心，数学系，LMRS UMR 6085）
el Houcein el Abdalaoui, Mahesh Nerurkar
备注：
23 pages. We present an unpublished theorem of Veech which is closely related to our main result with complete proof. The notion of Quasi-factor plays a key role. In this version, we give more details on the proof of our main result. Comments and suggestions are welcome
It is shown that Sarnak's Möbius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the Möbius function and its square. Then a computation of W. Veech, followed by an argument using the notion of `affinity between measures', (or the so-called `Hellinger method'), completes the proof. We further present an unpublished theorem of Veech which is closely related to our main result. This theorem asserts, if for any probability measure in the closure of the Cesaro averages of the Dirac measure on the shift of the Möbius function, the first projection is in the orthocomplement of its Pinsker algebra then Sarnak Möbius disjointness conjecture holds. Among other consequences, we obtain a simple proof of Matom{ä}ki-Radziwiłl-Tao's theorem and Matom{ä}ki-Radziwiłl's theorem on the correlations of order two of the Liouville function.
On spectral disjointness of powers for rank-one transformations and Möbius orthogonality
本文研究秩一变换的幂的谱不交性。对于一个大类的秩一的建设，包括那些切割和堆叠参数有界的，和其他的例子，如刚性广义Chacon的地图和Katok的地图，我们证明了不同的正功率的变换是成对的频谱不相交的连续部分的频谱。我们的证明涉及到在{U_T^k：k in Z}的弱闭包中，算子U_T的“足够多”解析函数的存在性.然后，我们应用这些不相交的结果证明Sarnak猜想（可能非唯一遍历）与这些秩一结构相关的符号模型：在这些模型中实现的所有序列都正交于默比乌斯函数。
