Renormalization in piecewise isometric dynamical systems

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A collection of objects whose changes are governed by a time-independent update rule is called a dynamical system. Examples are pervasive in the sciences, including the motion of celestial bodies, weather patterns, the behavior of fluids moving through a pipe, and chemical processes. A basic question in the subject is to determine the behavior of such a system. So called hyperbolic dynamical systems are a prime example of success in the field. These systems admit a special kind of expansion and contraction, which make available certain mathematical methods of analyzing the behavior of these systems. This project centers on understanding systems at the opposite extreme, piecewise isometric systems, which admit no contraction or expansion and thus force different approaches to be used to understand these systems. These systems are typically understood through renormalization. Renormalization is a method of looking more and more closely at, or zooming in on, repeated behavior. Renormalization is a major part of this project and the PI will develop new renormalization methods and improve existing methods to understand these systems. A better understanding of these systems will have broad consequences for our understanding of dynamical systems as a subject. In addition to the research aims of the project, the PI will work with students, providing mentoring and training in mathematical research.The PI will make contributions toward the use of renormalization to understand dynamical phenomena in piecewise isometric dynamical systems and related systems. Systems of interest include horocyclic flows on hyperbolic surfaces, geodesic flows on flat manifolds, and piecewise isometric systems. Many of these systems arise naturally through connections with low-dimensional topology and geometry. A piecewise isometry is formed by cutting a metric space into pieces and applying an isometry to each piece so as to reassemble the whole space. Examples include interval exchange transformations (IETs), where the space is an interval, which is cut into finitely many subintervals. The theory of IETs is quite well developed and stands as a testament to the power of renormalization methods, but in contrast, many related systems are only poorly understood. A primary goal is to extend the applicability of renormalization methods to wider classes of systems including polygon exchange transformations, and infinite interval exchange transformations. New dynamical phenomena will be discovered and rigorously studied using the renormalization methods developed by the PI. Results produced by research in this proposal will widely disseminated through publication in research journals and through conference presentations.