Quantitative characteristics of the hyperbolic sets arising in conservative dynamics, celestial mechanics, and spectral theory

保守动力学、天体力学和谱理论中出现的双曲集的定量特征

• 批准号: 0901627
• 负责人: Anton Gorodetski
• 金额: $ 26.48万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901627&HistoricalAwards=false
• 关键词: Quantitative; characteristics; hyperbolic; sets; arising

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The project focuses on quantitative properties (such as Hausdorff dimension and thickness) of hyperbolic sets that appear in conservative dynamics, celestial mechanics, and spectral theory. The first part of the project deals with hyperbolic sets generated by a conservative homoclinic bifurcation. These hyperbolic sets have large Hausdorff dimension, and that leads to a series of applications. In particular, this phenomenon can be applied to study the oscillatory motions in the three-body problem. A motion in a three-body problem is called "oscillatory" if for an unbounded increasing sequence of times the diameter of the system appears as bounded, while for another unbounded sequence of times the diameter of the system goes to infinity. Oscillatory motions are directly related to homoclinic pictures and invariant hyperbolic sets. The goal of this part of the project is to demonstrate the existence of homoclinic bifurcations in some restricted versions of the three-body problem and to show that the set of oscillatory motions has full Hausdorff dimension for many parameter values. Another part of the project is an application of the theory of hyperbolic dynamical systems to spectral theory. Namely, using the hyperbolicity of the so-called trace map and estimating some quantitative characteristics of its invariant sets, one can derive new properties of the spectrum of the discrete Schrodinger operator with Fibonacci potential. That will provide new insights on the properties of quasicrystals. Many of the problems considered in the project were initially formulated by physicists, and results may have applications to the theory of quasicrystals, to comet dynamics within the solar system, and to a number of other problems in physics, chemistry, and astronomy. Various problems closely related to the project will be suggested to graduate and undergraduate students at UC-Irvine, thereby initiating them into or increasing their involvement in scientific activities. The principal investigator considers such educational and training aspects to be central to the project.

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。该项目专注于守恒动力学、天体力学和谱理论中出现的双曲集的定量性质(如Hausdorff维度和厚度)。该项目的第一部分处理由保守同宿分支产生的双曲集。这些双曲集具有很大的Hausdorff维数，这导致了一系列的应用。特别是，这一现象可用于研究三体问题中的振荡运动。在三体问题中，如果对于一个无限递增的时间序列，系统的直径看起来是有界的，而对于另一个无限的时间序列，系统的直径是无穷大的，那么这个运动称为“振荡”。振荡运动与同宿图和不变双曲集直接相关。这一部分的目的是证明三体问题的某些受限形式的同宿分支的存在性，并证明对于许多参数值，振荡运动集具有全Hausdorff维。该项目的另一个部分是双曲动力系统理论在光谱理论中的应用。也就是说，利用所谓迹映射的双曲性和估计其不变集的一些定量特征，可以得到具有斐波那契势的离散薛定谔算子的谱的新性质。这将为准晶的性质提供新的见解。该项目中考虑的许多问题最初是由物理学家提出的，结果可能适用于准晶理论、太阳系内的彗星动力学以及物理、化学和天文学中的其他一些问题。将向加州大学欧文分校的研究生和本科生提出与该项目密切相关的各种问题，从而启动或增加他们对科学活动的参与。首席调查员认为，这些教育和培训方面对该项目至关重要。