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

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。该项目重点研究出现在保守动力学、天体力学和谱理论中的双曲集的定量性质（如Hausdorff维数和厚度）。项目的第一部分处理由保守同斜分岔生成的双曲集。这些双曲集有很大的豪斯多夫维数，这导致了一系列的应用。特别地，这种现象可以应用于研究三体问题中的振荡运动。一个三体问题中的运动被称为“振荡”，如果对于一个无界的增加序列系统的直径是有界的，而对于另一个无界的增加序列系统的直径是无穷大的。振荡运动与同斜象和不变双曲集直接相关。本课题的目标是证明在某些限制版本的三体问题中存在同斜分岔，并证明振荡运动集对于许多参数值具有完整的豪斯多夫维数。项目的另一部分是双曲动力系统理论在谱理论中的应用。也就是说，利用所谓迹映射的双曲性并估计其不变集的一些定量特征，可以推导出具有斐波那契势的离散薛定谔算子谱的新性质。这将为准晶体的性质提供新的见解。该项目中考虑的许多问题最初都是由物理学家提出的，其结果可能应用于准晶体理论、太阳系内的彗星动力学以及物理、化学和天文学中的许多其他问题。与项目密切相关的各种问题将被建议给加州大学欧文分校的研究生和本科生，从而引导他们进入或增加他们对科学活动的参与。首席研究员认为这些教育和培训方面是项目的核心。

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.