On spectral disjointness of powers for rank-one transformations and Möbius orthogonality
On spectral disjointness of powers for rank-one transformations and Möbius orthogonality
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DOI：
10.1016/j.jfa.2013.09.005
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发表时间：
2013-01
影响因子：
1.7
通讯作者：
E. Abdalaoui;M. Lemanczyk;T. Rue
中科院分区：
文献类型：
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作者：
E. Abdalaoui;M. Lemanczyk;T. Rue
我们研究一阶变换的幂的谱不交性。对于一大类一阶结构，包括切割和堆积参数有界的结构，以及其他例子，如刚性广义ChaconʼS映射和KatokʼS映射，我们证明了变换的不同正幂在谱的连续部分是两两谱不交的。我们的证明涉及到在{U T k：K∈Z}的弱闭包下，算子U T的“足够多”解析函数的存在性。然后，我们利用这些不交结果证明了与这些一阶结构有关的符号模型(可能不是唯一的遍历的)的SarnakʼS猜想：在这些模型中实现的所有序列都与Möbius函数正交。
We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chaconʼs maps and Katokʼs map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of {U T k: k∈ Z}, of “sufficiently many” analytic functions of the operator U T. Then we apply these disjointness results to prove Sarnakʼs conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Möbius function.