Ergodicity, Rigidity, and the Interplay Between Chaotic and Regular Dynamics

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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 Newtonian forces controlling mechanical 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 research will address the theoretical mechanisms behind chaotic motion in broad classes of dynamical systems, which include systems of both a physical and geometric nature. Based on previous work of the principal investigator and her collaborators, the interplay between a numerical invariant called entropy and chaotic motion will be further understood. On the flip side, the principal investigator proposes several problems connecting entropy and related invariants with a phenomenon called rigidity. Rigidity occurs when quantitatively small changes to the rules guiding a system force fundamental qualitative changes in the resulting dynamics. Identifying the rigid systems is a first step toward classification of certain peculiar dynamical behaviors observed in physical and geometric systems. An important aspect of the project is to further interaction between mathematical and adjacent scientific communities, such as physics. The principal investigator has already collaborated in questions surrounding the design of particle accelerators and is currently collaborating with a physicist studying the quantum dynamics behind the emergence of black holes. Furthermore, the principal investigator has given several public lectures on dynamics and has written in the popular press about the work of mathematicians. She proposes to expand these activities in the coming years.Dynamics is the study of systems (for example, a state space for a physical process) that evolve over time according to a deterministic set of rules. Well-studied classes of such dynamical systems include the so-called hyperbolic systems, which display chaotic, unpredictable features at every point, and KAM systems, which have stable regions of regular motion. The partially hyperbolic systems are a more general class of dynamical systems than the hyperbolic class and include systems that combine hyperbolicity in some directions with KAM behavior in other directions. Partially hyperbolic systems occur widely in dynamical systems arising in physics; for example, planetary motion usually contains partially hyperbolic sub-dynamics, and the effective construction of particle accelerators (used in biological imaging, as well as theoretical physics) requires a detailed understanding of both KAM and partially hyperbolic dynamics. The principal investigator has a well-developed research plan of over 15 years studying partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability. The impacts of this research will be seen in future applications to systems of a concrete origin, in biology, physics and engineering. The principal investigator is currently collaborating with the particle accelerator group at Fermilab to explore some of these potential applications. The research supported by this award is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed in the past 40 years. In particular the principal investigator proposes to study: ergodic properties of conservative partially hyperbolic diffeomorphisms; actions of large collections of diffeomorphisms and embeddings on manifolds; and rigidity phenomena in actions of groups. A highlight of the proposed research is to investigate the interaction between hyperbolicity and KAM phenomena.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.