Newhouse Phenomena in Celestial Mechanics and Spectral Theory

事实上，一个小的交集（零测度，无处稠密）集可以是持久的，即不能被一个小的平移所破坏。（或更一般的集合扰动类）在动力系统中有许多有趣而重要的结果（同宿切线），数论（由有界类型的连续分数表示的无理数集），谱理论（具有可分离势的算子）和其他数学领域。 最著名的应用之一是现在所谓的纽豪斯现象-存在一个二维表面的光滑同胚持续同宿切线的开放集。 这个项目的目标是提高我们对保守和耗散动力系统的康托集和纽豪斯现象的和/相交的一般问题的理解，并回到原来的问题（例如从天体力学），以及考虑一些新的模型（例如从谱理论），这些结果可以应用。 指导学生是该项目的重要组成部分。 许多问题密切相关的拟议项目将建议研究生和本科生发起和增加他们的参与到科学活动和研究。更具体地说，PI计划研究纽豪斯现象的三个机构的问题（如Sitnikov问题，圆形限制性三体问题，共线限制性三体问题，以及非限制性版本），并将保守同宿分支的结果应用于无穷远动力学问题。 特别是，他打算研究的一套振荡运动在这些模型，并在同一时间表明，一套稳定的运动，开始在一些给定的有界域可以无界。 此外，还将研究具有可分离势的二维晶格上的薛定谔算子的谱结构（特别是离散和连续版本的平方斐波那契汉密尔顿）。 在有限参数族中康托集的和的问题将被考虑，与著名的帕利斯猜想的结构上的仿射康托集的总和在地平线上。 纽豪斯现象及其在台球地图中的后果也将被研究。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.