Stabilization Methods for Dynamical Systems

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The study of dynamical systems concerns the structure of systems which evolve in time. Their study connects with several areas within mathematics, as well as physics and other fields of science. This project is focused on groups of symmetries of dynamical systems, exploring how properties of the underlying system manifest in, and constrain, the structure of these symmetry groups. A key class of systems are the hyperbolic homeomorphisms, and the core of the project concerns a new stabilized framework for studying the collection of automorphisms of these systems. Symbolic systems, including shifts of finite type, also play a key role, and have important connections to other areas such as ergodic theory and information theory, and their symmetry groups exhibit a wide range of complexity. This project will advance the field by investigating new techniques and invariants for classifying symmetries of systems through the use of stabilization. The project also provides opportunities for graduate students to conduct research. The project is aimed at a notion of reconstruction: the extent to which the structure of symmetry groups, and their stabilized counterparts, can be used to reconstruct various dynamical properties of the underlying system. The PI intends to further develop and apply various stabilized invariants, notable local entropy, to the study of such automorphism groups with the goal of recovering dynamical invariants, such as topological entropy and zeta functions. Such invariants have been used successfully in the symbolic setting, and one aim of the project is to extend this to certain smooth settings. One goal of the project is to study the question of whether the stabilized automorphism group of a shift of finite type is a complete invariant up to topological conjugacy. In addition, the project explores the extent to which stabilized results, and invariants derived from there, might be used to analyze the classical automorphism groups.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.