UNIVERSITY OF CALIFORNIA, IRVINE
Grant: $264,793 - National Science Foundation - Jul. 2, 2009
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Award Description: Dynamically defined Cantor sets play a fundamental role in Dynamical Systems. The proposed project focuses on two topics related to the problems where dynamically defined Cantor sets appear: conservative Newhouse phenomena in celestial mechanics, and the dynamical properties of the Trace Map that describe the spectral properties of the discrete Schr?odinger operator with Fibonacci potential.First part of the proposed project deals with oscillatory motions in the three body problems. 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. Amazingly enough,oscillatory motions are directly related to homoclinic tangencies and Newhouse phenomena. The goal of the first part of the project is to show how Newhouse phenomena appear in the three body problems, and to derive some consequences regarding oscillatory motions that are important and interesting within the context of celestial mechanics and, more generally, in Hamiltonian dynamics. The second part of the project is an interesting application of the theory of hyperbolic dynamical systems to the spectral theory. Namely, we are going to show that so called Trace Map is hyperbolic, and use this result to establish the transport properties of the Fibonacci Hamiltonian. Intellectual Merit The first part of the project will lead to a much better understanding of the structure of the set of oscillatory motions in the three body problem, and, therefore, will make one more step along the way initiated by Chazy almost one hundred years ago. As a result of the second part of the project the Hausdor? dimension of the spectrum of the discrete Schr?odinger operator with Fibonacci potential will be evaluated, and transport properties for small coupling will be established. Applications of these results to some other physical problems will be also considered. Broader Impacts The PI plans to suggest numerous problems closely related to the proposed project, to graduate and undergraduate students in UC Irvine initiating and increasing their involvement to scientific activities. Many of the problems considered in the project were initially formulated by physicists, and results may have applications in plasma physics, particle dynamics in accelerators, comet dynamics in solar system, quasicrystals, and other problems in physics, chemistry, and astronomy. A large and increasing amount of examples appear showing that theory of Dynamical Systems can be used by research groups in Applied Mathematics and in Mathematical Physics(traditionally strong in Southern California) to produce beautiful results. If supported, this project will essentially increase the productive interactions between people from different but related branches of mathematics.
Project Description: Dynamically defined Cantor sets play a fundamental role in Dynamical Systems. The proposed project focuses on two topics related to the problems where dynamically defined Cantor sets appear: conservative Newhouse phenomena in celestial mechanics, and the dynamical properties of the Trace Map that describe the spectral properties of the discrete Schr?odinger operator with Fibonacci potential.First part of the proposed project deals with oscillatory motions in the three body problems. 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. Amazingly enough,oscillatory motions are directly related to homoclinic tangencies and Newhouse phenomena. The goal of the first part of the project is to show how Newhouse phenomena appear in the three body problems, and to derive some consequences regarding oscillatory motions that are important and interesting within the context of celestial mechanics and, more generally, in Hamiltonian dynamics. The second part of the project is an interesting application of the theory of hyperbolic dynamical systems to the spectral theory. Namely, we are going to show that so called Trace Map is hyperbolic, and use this result to establish the transport properties of the Fibonacci Hamiltonian. Intellectual Merit The first part of the project will lead to a much better understanding of the structure of the set of oscillatory motions in the three body problem, and, therefore, will make one more step along the way initiated by Chazy almost one hundred years ago. As a result of the second part of the project the Hausdor? dimension of the spectrum of the discrete Schr?odinger operator with Fibonacci potential will be evaluated, and transport properties for small coupling will be established. Applications of these results to some other physical problems will be al
Jobs Summary: Created: N/A Retained: N/A (Total jobs reported: 0)
Project Status: Less Than 50% Completed
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