Ergodic Properties of Smooth Systems on Manifolds

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This research project concerns the chaotic properties of smooth dynamical systems. Smooth dynamical systems may exhibit chaotic behavior, this is where the future evolution of the systems is strongly independent of its present state and the evolution behaves like a sequence of independent coin tosses. The principal investigator plans to study such chaotic behavior for a wide class of natural dynamical systems. The developed methods will result in progress in the understanding of fundamental dynamical phenomena, with possible applications to other mathematical fields such as geometry and number theory, and also to the natural sciences. The principal investigator also plans to be involved in synergistic activities including organizing conferences and working with graduate students and postdocs.Chaotic properties of (uniformly) hyperbolic systems are by now well understood. Much less is known for partially hyperbolic systems, especially those with non-trivial (fast) growth on the center space. The principal investigator plans to create a general framework for studying K and Bernoulli properties for partially hyperbolic systems, and also plans to study mixing and spectral properties of parabolic systems, trying to further develop a general theory for systems of polynomial orbit growth.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.