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

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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.