Newhouse Phenomena in Celestial Mechanics and Spectral Theory

University Of California-Irvine, Irvine CA

Investigators

Abstract

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.