论文标题
弱驯服的系统，其特征和应用
Weakly tame systems, their characterizations and application
论文作者
论文摘要
We explore the notion of discrete spectrum and its various characterizations for ergodic measure-preserving actions of an amenable group on a compact metric space.我们介绍了一个“弱感知”的概念，这是E. Glasner引入的“驯服”概念的衡量理论版本，该概念是根据A.Köhler的工作[A. Köhler引入了这一概念，并将此类系统称为“常规”。 Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of 'witness of irregularity' which is based on up-crossings.然后，我们确定强大的Veech系统是驯服的。特别是，对于任何符合的组$ t $，在\ mathbb {k}（t）$ in \ mathbb {k}（t）$的轨道闭合上的流量是驯服的。 Thus Sarnak's Möbius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi-Ramachandra 1976's theorem on the Mertens function in short intervals.我们进一步提高了Motohashi-Ramachandra在Chowla猜想下的$ 1/2 $。
We explore the notion of discrete spectrum and its various characterizations for ergodic measure-preserving actions of an amenable group on a compact metric space. We introduce a notion of 'weak-tameness', which is a measure-theoretic version of a notion of `tameness' introduced by E. Glasner, based on the work of A. Köhler [A. Köhler introduced this notion and call such systems "regular".], and characterize such topological dynamical systems as systems for which every invariant measure has a discrete spectrum. Using the work of M. Talagrand, we also characterize weakly tame as well as tame systems in terms of the notion of 'witness of irregularity' which is based on up-crossings. Then we establish that strong Veech systems are tame. In particular, for any amenable group $T$, the flow on the orbit closure of the translates of a `Veech function' $f\in \mathbb{K}(T)$ is tame. Thus Sarnak's Möbius orthogonality conjecture holds for this flow and as a consequence, we obtain an improvement of Motohashi-Ramachandra 1976's theorem on the Mertens function in short intervals. We further improve Motohashi-Ramachandra's bound to $1/2$ under Chowla conjecture.
