Rigid Structures and Statistical Properties of Smooth Systems

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The aim of this project is to discover new phenomena in the area of dynamical systems. Dynamical systems ("dynamics," for short) is the study of motion, and in particular motion that is dictated by an unchanging set of rules, such as the forces controlling planetary motion. Well-known experimental phenomena in dynamics such as chaotic trajectories combined with stable motion have been observed experimentally but are far from being fully understood from a theoretical perspective. The project will address several themes. The first is the stability of dynamical systems - that is, how small perturbations in initial conditions and even in the rules themselves affect the outcome of the evolution. Understanding robust mechanisms for stability is a fundamental pursuit, and the investigator has already discovered several novel such mechanisms. The second theme is genericity, or loosely to understand what dynamical features are present in a typical system. The third theme is rigidity, the study of symmetries of dynamical systems and those systems with optimal symmetries, the so-called ideal crystals of dynamics. An important aspect of the project is to further interaction between mathematical and adjacent scientific communities, such as physics. The PI has already collaborated on questions surrounding the design of particle accelerators and is currently collaborating with a physicist on studying the quantum dynamics behind the emergence of black holes. Furthermore, the PI has given several public lectures on dynamics and has written in the popular press about the work of mathematicians. The PI will expand these activities in the coming years. The project provides research training opportunities for undergraduate and graduate students.This project considers questions in smooth dynamical systems all the way from a general perspective, in particular those about genericity of certain foliation dynamics, to a local one, focused on the rigidity of specific families of group actions. These questions are motivated by well-known conjectures, but also by the desire to discover and explore new dynamical phenomena. The first circle of questions centers around Boltzmann’s original ergodic hypothesis as well as the modern and related conjectures of Pugh and Shub about stable ergodicity. The basic question they address is when one might expect a dynamical system to be ergodic. An important question that remains open is the symplectic version of the C1 Pugh-Shub conjecture, which the investigator will attack. The strategy to prove this conjecture involves interesting and timely aspects of the study of hyperbolic and partially hyperbolic dynamics and expanding foliations. A second project investigates the topological and statistical properties of the unstable foliations of partially hyperbolic systems, and in particular Anosov diffeomorphisms with a partially hyperbolic splitting. A third project concerns the rigidity properties of partially hyperbolic abelian actions. Here the action of the su-holonomy group plays an important role: in the actions considered, the joint action of the ambient, partially hyperbolic dynamics and the su-holonomy group are constrained by certain solvable groups for which known rigidity results described above hold.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.