The proposed research extends two aspects of the principal investigator's previous research program in smooth dynamics. In the first, her work with collaborators has led recently to a proof of a "Boltzmann Ergodic Hypothesis" for partially hyperbolic systems: the typical conservative partially hyperbolic system is ergodic. In other recent work, the principal investigator and her collaborators have shown that typical conservative diffeomorphisms with minimal differentiability have trivial centralizers; that is, they have no smooth symmetries. The project has two main components: (1) extending the previous work of the principal investigator and her collaborators on the ergodicity of partially hyperbolic diffeomorphisms to a broader context, including dissipative systems and nonuniformly partially hyperbolic systems; (2) a study of the dynamical properties of smooth group actions on manifolds from the perspectives of genericity and rigidity. For one example, the principal investigator proposes to extend previous work with her collaborators in the conservative setting to show that the typical diffeomorphisms with minimal differentiability have trivial centralizers.
In the past few decades, the topic of partially hyperbolic dynamical systems has emerged as one main direction in which the theory of complicated ("chaotic") dynamical systems has extended beyond the classical setting of hyperbolic dynamics. This research has been driven in part by the realization that many practical applications of dynamical systems to experimental phenomena require a much broader theory. In a parallel development, the typical (or generic) properties of smooth systems have begun to emerge, as increasingly sophisticated perturbation techniques have been developed. The principal investigator's research over the last decade has focused on the fundamental qualitative features -- e.g., ergodicity (statistical "chaos"), lack of symmetries, and other hallmarks of orbit complexity -- that might be displayed by a typical smooth dynamical system. The proposed research will make foundational advances in understanding when to expect chaos in a given family of systems, such as those that arise in practice in predicting global weather patterns or the trajectories of extraterrestrial bodies. The material resulting from this research proposal will be widely disseminated, through talks at conferences, online preprint servers, and publication in scholarly journals. A significant component of this grant is devoted to graduate-student training on the Ph.D. level.
