论文标题
由萨尔纳克（Sarnak）的猜想，第二部分的振荡命令
Orders of Oscillation Motivated by Sarnak's Conjecture, Part II
论文作者
论文摘要
我已经调查了由萨尔纳克（Sarnak）在〜\ cite {jpams}中的猜想所激发的振荡序列的顺序，并证明了$ d $的振荡序列与$ d $ torus上的远端流相结合。后果之一是，算术意义上的$ d $的振荡顺序是与$ d $ -torus上零拓扑熵的仿射流线性脱节。在本文中，我将将这些结果扩展到$ d $ -torus上的多项式偏斜产品，也就是说，给定$ d $ -torus上的多项式偏斜产品，有一个积极的整数$ m $，以便任何$ m $ $ m $ $ m $的振动序列都是线性地脱离了这种polynomial skew skew skew skew skew skew。特别是，当所有多项式仅取决于第一个变量时，我的振荡顺序$ m = d+k-1 $是与$ d $ torus上的所有多项式偏斜产品线性分离，该$ d $ torus具有多项式学位的程度少于或等于$ k $。后果之一是流量的线性脱节，这是$ d $ torus的自动形态，具有特征值$ 1 $的绝对值加上多项式矢量，并且在算术意义上是$ M $的振荡序列。此外，我将证明，$ d $的振荡顺序是与最小的平均可吸引人和最小的准准盘频谱$ d $ d $ flow的最小值差异。最后，我从论文〜\ cite {aj}中定义并给出了一些chowla序列的示例。
I have investigated orders of oscillating sequences motivated by Sarnak's conjecture in~\cite{JPAMS} and proved that an oscillating sequence of order $d$ is linearly disjoint from affine distal flows on the $d$-torus. One of the consequences is that an oscillating sequence of order $d$ in the arithmetic sense is linearly disjoint from affine flows with zero topological entropy on the $d$-torus. In this paper, I will extend these results to polynomial skew products on the $d$-torus, that is, given a polynomial skew product on the $d$-torus, there is a positive integer $m$ such that any oscillating sequence of order $m$ is linearly disjoint from this polynomial skew product. In particular, when all polynomials depend only on the first variable, I have that an oscillating sequence of order $m=d+k-1$ is linearly disjoint from all polynomial skew products on the $d$-torus with polynomials of degree less than or equal to $k$. One of the consequences is the linear disjointness for flows which are automorphisms of the $d$-torus with absolute values of eigenvalues $1$ plus a polynomial vector and oscillating sequences of order $m$ in the arithmetic sense. Furthermore, I will prove that an oscillating sequence of order $d$ is linearly disjoint from minimal mean attractable and minimal quasi-discrete spectrum of order $d$ flows. Finally, I define and give some examples of Chowla sequences from our paper~\cite{AJ}.
