Ergodic Properties of Smooth Systems on Manifolds

This project focuses on the chaotic properties of smooth dynamical systems, a very active area of current research in mathematical analysis, with connections and applications across the mathematical sciences. Chaotic behavior in dynamical systems is characterized by sensitive dependence on initial conditions, which occurs when small changes in a system’s present state may produce large fluctuations in its future state. This occurs in many natural systems, such as the human heart, fluid flows, and global weather patterns. The PI plans to develop a general framework and new approaches for understanding chaotic behavior in a large class of dynamical systems. Activities in the project will lead to progress in our understanding of fundamental dynamical phenomena, with possible consequences and applications in other mathematical fields, such as geometry and number theory, and other scientific areas, such as physics and economics. The project will also involve the training of several students and postdocs. This project is part of a program of research studying ergodic and statistical properties of smooth systems on manifolds and their interactions with geometry and number theory, with three main areas of focus. One area concerns some classical properties that are expected to hold for chaotic systems: K and Bernoulli properties, quantitative mixing (and higher order mixing) and limit theorems. The PI will investigate relations between these properties and their appearance for partially hyperbolic (or non-uniformly partially hyperbolic) systems, and continue developing a geometric framework for problems related to quantitative mixing and the Bernoulli property. The PI also plans to construct examples of exotic dynamical systems with new ergodic and statistical behavior. A second research direction relates to recent developments on parabolic systems, of not necessarily algebraic origin. Despite recent progress, many fundamental questions for such systems are still open, for instance, the Rokhlin problem on higher order mixing. The PI will build on techniques from his earlier work and try to develop a general theory for ergodic properties of parabolic systems. A third part of the project will continue the PI’s research on sparse equidistribution problems, using methods from dynamics (such as quantitative equidistribution and mixing) and analytic number theory (such as sieve methods and exponential sums). 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.