Stability, typicality and rigidity in smooth dynamics

University Of Chicago, Chicago IL

Investigators

Abstract

This project explores why some systems behave predictably while others react chaotically to even the slightest nudge — a question with consequences for everything from weather forecasting to data security. By uncovering the mathematical rules that govern the coexistence of order and chaos, the work strengthens the nation’s scientific knowledge base and supports future technological innovation. Findings will be shared widely through public lectures, podcasts, and a forthcoming graduate-level book, “Dynamics, Rigidity, and Geometry,” making cutting-edge ideas accessible to students, educators, and lifelong learners. The research tackles three intertwined themes in smooth dynamics: 1. Stability. Identify mechanisms that keep chaotic or regular behavior intact under small perturbations. A major target is the symplectic C1 ergodic hypothesis, with new perturbation and blender techniques expected to yield density results for stably ergodic symplectomorphisms. 2. Typicality. Determine which dynamical properties occur for “most’’ systems. Work on expanding and unstable foliations aims to prove openness and density of minimal strong foliations and to establish uniqueness of associated equilibrium (u-Gibbs) measures, with consequences for their higher statistical properties. 3. Rigidity. Classify maps that possess large symmetry groups. The project’s “affine centralizer program” links the algebraic size of a diffeomorphism’s centralizer to its long-term dynamics, producing new classifications for partially hyperbolic actions on nilmanifolds and other homogeneous spaces. Methods combine C1 perturbation theory, entropy-preserving linearization, blender and super-blender constructions, higher-rank algebraic actions, and modern measure-rigidity tools. Expected outcomes include: definitive progress on the symplectic ergodic hypothesis; a deeper understanding of how geometric structures dictate ergodic and mixing behavior; and refined criteria for when chaotic dynamics are stable or typical. Results will appear in refereed journals and preprint archives and will be integrated into the PI’s book, providing the community with a unified exposition of new techniques and discoveries while training graduate students and postdoctoral scholars in a rapidly advancing field. 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.