RUI: Generalized Gauss Maps and Divergent Orbits

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The main focus of this project is on the development of technology that predicts the behavior of certain mathematically defined dynamical systems that are platonic idealizations of a broad class of mathematical models used to describe everyday physical processes of the real world. Apart from helping us understand natural phenomena amid random fluctuations, this technology is also becoming increasingly necessary for understanding complex systems built by humans to meet the demands of modern society. The principal investigator will participate in various workshops and conferences around the world to disseminate the most recent discoveries of his research program and to keep up with latest developments in the field. These trips also provide the opportunity for PI to work with his collaborators. PI will also invite some of his international collaborators to the U.S. to work with him on their joint projects. This project addresses a long-standing problem to generalize the well-known combinatorial description of geodesics on the modular surface whose dynamical behavior is intimately tied to the theory of approximation of irrational numbers by rationals. The principal investigator proposes several generalizations of the seamless integration of number theory and dynamics to higher dimensions using several well-chosen Poincare sections of homogeneous flows to replace the all-powerful Gauss continued fraction algorithm, and introducing Farey tilings to take the role of continued fractions and provide a combinatorial description of the special linear group under its full diagonal subgroup. These generalizations germinated in the principal investigators previous work on the determination of the Hausdorff dimension of the set of singular vectors and promise to lead to new insights of fundamental significance in simultaneous approximation theory.