Applications of Descriptive Set Theory in Ergodic Theory and Smooth Dynamical Systems

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Models of physical phenomena are often formulated as dynamical systems, rules that specify how quantities evolve in time. There are many different types of dynamical systems, and one important mathematical question is to classify the statistical or qualitative behavior of these systems. This project uses descriptive set theory to show what kinds of classification are possible and to establish the mathematical limitations of general classifications. The project involves graduate students.Classification of dynamical systems can be approached by classifying the statistical and qualitative behavior of diffeomorphisms of compact manifolds and establishing criteria for deciding when time forwards is isomorphic to time backwards (in the appropriate category). These questions are closely related to which abstract measure-preserving transformations are isomorphic to volume-preserving diffeomorphisms of compact manifolds. This project studies when classifications are possible with inherently countable methods and explores methods for realizing abstract transformations as diffeomorphisms.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.