Newhouse Phenomena in Celestial Mechanics and Spectral Theory

在动力系统(同宿切)、数论(由有界型的连续分式表示的无理数的集合)、谱理论(具有可分离势的算子)和其他数学领域中，小(零度量，无处稠密)集合的交集可以是持久的，即不能被小平移(或更一般的集合的扰动)摧毁的事实在动力系统(同宿切线)、数论(由有界型的连续分式表示的无理数的集合)和其他数学领域有许多有趣和重要的结果。其中最著名的应用之一就是现在所说的纽豪斯现象--具有持久同宿切的二维曲面的光滑微分同胚开集的存在性。这个项目的目的是提高我们对保守和耗散动力系统的Cantor集的和/交和/交和/和的一般问题的理解，并回到原始问题(例如从天体力学)，以及考虑一些新的模型(例如从谱理论)，这些结果可以在那里应用。辅导学生是该项目的重要组成部分。许多与拟议项目密切相关的问题将被建议给研究生和本科生，以启动和增加他们对科学活动和研究的参与。更具体地说，PI计划研究三体问题(如Sitnikov问题、圆形限制三体问题、共线限制三体问题以及非限制版本)中的Newhouse现象，并将保守同宿分支的结果应用于无限大附近的动力学问题。特别是，他打算研究这些模型中的振荡运动集合，同时证明在某个给定的有界域中开始的稳定运动集合可以是无界的。此外，还将研究具有可分势(特别是离散和连续形式的平方斐波那契哈密顿量)的二维晶格上薛定谔算符的谱结构。利用著名的Palis关于水平线上仿射Cantor集和结构的猜想，我们将考虑有限参数族内的Cantor集的和的问题。此外，还将研究台球地图中的纽豪斯现象及其后果。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。

The fact that an intersection of small (zero measure, nowhere dense) sets can be persistent, i.e. cannot be destroyed by a small translation (or even more general class of perturbations of the sets) has many interesting and important consequences in dynamical systems (homoclinic tangencies), number theory (sets of irrational numbers represented by continuous fractions of bounded type), spectral theory (operators with separable potentials), and other areas of mathematics. One of the most famous applications is what nowadays called Newhouse phenomenon - existence of open sets of smooth diffeomorphisms of a two-dimensional surface with persistent homoclinic tangencies. The goal of this project is to improve our understanding of general questions on sums/intersections of Cantor sets and Newhouse phenomena for conservative and dissipative dynamical systems, and also to go back to the original problems (e.g. from celestial mechanics), as well as to consider some new models (e.g. from spectral theory), where these results can be applied. Mentoring students is an essential part of the project. Numerous problems closely related to the proposed project will be suggested to graduate and undergraduate students initiating and increasing their involvement into scientific activities and research.More specifically, the PI plans to study Newhouse phenomena in the three body problems (such as Sitnikov problem, circular restricted three body problem, and collinear restricted three body problem, as well as in non-restricted versions), and to apply the results on conservative homoclinic bifurcations to the questions on dynamics near infinity. In particular, he intends to study the set of oscillatory motions in these models, and at the same time to show that the set of stable motions that start in some given bounded domain can be unbounded. Also, the structure of spectra of Schrodinger operators on two-dimensional lattice with separable potentials (in particular, Square Fibonacci Hamiltonian both in discrete and continuous version) will be studied. Questions on sums of Cantor sets within a finite-parameter family will be considered, with the famous Palis' Conjecture on the structure of sums of affine Cantor sets on a horizon. Newhouse phenomena and their consequences in billiard maps will be studied as well.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.