Dynamics and Kleinian Groups

动力学和克莱尼群

• 批准号: 1900101
• 负责人: Hee Oh
• 金额: $ 53.1万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900101&HistoricalAwards=false
• 关键词: Dynamics; Kleinian; Groups

The three-dimensional hyperbolic space can be modeled by the upper half space of the three-dimensional Euclidean space, where the shortest path between two points is either along a vertical line or a vertical semi-circle. The boundary of this space is the plane together with the point at infinity. Most often the trajectory of a collection of symmetries in this space accumulates on a fractal on the plane, the study of which is related to dynamics of flows on the hyperbolic manifold, given by the quotient of the hyperbolic three space by the symmetry group. One beautiful example of such fractals is the Apollonian gasket which is constructed by a theorem proven by the ancient Greek geometer Apollonius. Fractals are abundant in nature as can be seen in the coastlines, ocean waves, Romanesco cauliflowers, etc. A drop of oil in soy sauce creates a circle packing akin to the Apollonian gasket. This project is expected to be of value in interpreting the structure of fractals arising in nature.Whereas most of hyperbolic three manifolds have infinite volume, the dynamical rigidity properties of hyperbolic manifolds have been investigated mostly for finite volume cases. The project concerns with dynamics on infinite volume hyperbolic manifolds, which turn out to be quite a rich theory and has several deep and striking applications. This project involves new interactions between hyperbolic geometry, Kleinian groups, ergodic theory and number theory. Investigating this new territory requires many new ideas both from geometry and dynamics and opens up new interactions between these fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

三维双曲空间可以用三维欧几里德空间的上半空间来建模，其中两点之间的最短路径要么沿着垂直线，要么沿着垂直半圆。这个空间的边界是平面和无穷远处的点。最常见的情况是，这个空间中的一组对称的轨迹积累在平面上的一个分形图上，它的研究与双曲流形上的流动动力学有关，由双曲三个空间的对称群的商给出。这种分形图的一个美丽例子是阿波罗尼亚垫片，它是由古希腊几何学家阿波罗尼乌斯证明的一个定理构造的。在自然界中，可以看到海岸线、海浪、罗马内斯科花椰菜等丰富的分形物。酱油中的一滴油形成了一个类似阿波罗尼亚垫圈的圆形包装。由于大多数双曲流形都是无限体积的，双曲流形的动力学刚性性质主要是在有限体积的情况下研究的。该项目涉及无限体积双曲流形上的动力学，这是一个相当丰富的理论，并有几个深刻而引人注目的应用。这个项目涉及双曲几何、克莱因群、遍历理论和数论之间的新相互作用。调查这一新领域需要许多来自几何学和动力学的新想法，并开辟了这些领域之间的新互动。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Anosov groups: local mixing, counting and equidistribution

Anosov 群：局部混合、计数和均匀分布

• DOI: 10.2140/gt.2023.27.513
• 发表时间: 2023
• 期刊: Geometry & Topology
• 影响因子: 2
• 作者: Edwards, Samuel;Lee, Minju;Oh, Hee
• 通讯作者: Oh, Hee

Uniqueness of Conformal Measures and Local Mixing for Anosov Groups

Anosov群保形措施和局部混合的独特性

• DOI: 10.1307/mmj/20217222
• 发表时间: 2022
• 期刊: Michigan Mathematical Journal
• 影响因子: 0.9
• 作者: Edwards, Sam;Lee, Minju;Oh, Hee
• 通讯作者: Oh, Hee

Geodesic planes in geometrically finite acylindrical -manifolds

几何有限圆柱流形中的测地平面

• DOI: 10.1017/etds.2021.19
• 发表时间: 2022
• 期刊: Ergodic Theory and Dynamical Systems
• 影响因子: 0.9
• 作者: BENOIST, YVES;OH, HEE
• 通讯作者: OH, HEE

Isolations of geodesic planes in the frame bundle of a hyperbolic 3-manifold

双曲 3 流形的框架丛中测地平面的隔离

• DOI: 10.1112/s0010437x22007928
• 发表时间: 2023
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Mohammadi, Amir;Oh, Hee
• 通讯作者: Oh, Hee

Horospherical invariant measures and a rank dichotomy for Anosov groups

Anosov 群的星球不变测度和等级二分法

• DOI: 10.3934/jmd.2023009
• 发表时间: 2023
• 期刊: Journal of Modern Dynamics
• 影响因子: 1.1
• 作者: Landesberg, Or;Lee, Minju;Lindenstrauss, Elon;Oh, Hee
• 通讯作者: Oh, Hee