Combinatorial constructions in Smooth Ergodic Theory

平滑遍历理论中的组合构造

• 批准号: 405305501
• 负责人: Dr. Philipp Kunde
• 金额: --
• 依托单位: Bereich Differentialgleichungen und Dynamische Systeme (DD)
• 依托单位国家: 德国
• 项目类别: Research Fellowships
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/405305501?language=en
• 关键词: Combinatorial; constructions; Smooth; Ergodic; Theory

Historically motivated by problems in statistical mechanics Ergodic Theory examines statistical properties of dynamical systems. In particular, one is interested in the long-term behaviour of the system as well as the relationship between its time and space averages. One of the main questions in Ergodic Theory asks if there are smooth maps with specific ergodic properties. This is also the central question of this research project.One of the most powerful tools of constructing smooth diffeomorphisms with prescribed ergodic or topological properties is the so-called approximation by conjugation-method developed by D. Anosov and A. Katok which works on arbitrary smooth compact connected manifolds of dimension at least 2 admitting a non-trivial circle action. These diffeomorphisms are constructed as limits of conjugates of maps belonging to the circle action. In this research project we aim at the smooth realization of further ergodic as well as spectral properties. Moreover, we want to continue extending the approximation by conjugation-method to the real-analytic category.

遍历理论的历史动机是统计力学中的问题，遍历理论研究了动力系统的统计特性。特别是，人们对系统的长期行为以及其时间和空间平均值之间的关系感兴趣。遍历理论中的一个主要问题是，是否存在具有特定遍历性质的光滑映射。构造具有给定遍历性或拓扑性的光滑复同态的最有力的工具之一是D. Anosov和A. Katok工作在至少2维的任意光滑紧连通流形上，允许一个非平凡的圆作用。这些非同构被构造为属于圈作用的映射的共轭的极限。在这个研究项目中，我们的目标是顺利实现进一步遍历以及频谱特性。此外，我们还想继续将共轭近似推广到实解析范畴。

项目成果

Spectral disjointness of powers of diffeomorphisms with arbitrary Liouvillean rotation behavior

具有任意刘维尔旋转行为的微分同胚幂的谱不相交

• DOI: 10.4064/sm191202-31-8
• 发表时间: 2021
• 期刊: Studia Mathematica
• 影响因子: 0.8
• 作者: P. Kunde