A Comprehensive Program in Smooth Local and Global Rigidity

The mathematics of dynamical systems, which concerns models of time-evolving processes, is both a vital subfield of mathematical analysis, and plays a critical role in applications from weather prediction and power-grid stability to encrypted communications and spacecraft navigation. This project will develop novel analytical frameworks for studying rigidity phenomena in weakly chaotic dynamical systems, advancing both theoretical insights and practical applications. Rigidity in dynamical systems refers to the persistence of certain structures or behaviors under small perturbations. Understanding rigidity in weakly chaotic settings is critical: it can lead to more reliable climate forecasts, safer engineering designs, and new insights into the organization of complex networks. The project also includes an educational component: graduate mentoring, public seminars, and online tutorials to cultivate the next generation of U.S. researchers in STEM. This research project is organized in two branches. 1. Higher-rank actions via KAM theory. The project will extend the PI's prior work on smooth rigidity by applying Kolmogorov-Arnold-Moser (KAM) scheme to treat general higher-rank parabolic actions. The aim is to prove persistence of invariant structures under small perturbations, even in the absence of classical uniform hyperbolicity. 2. Smooth conjugacy in rank-one dynamics. The project will address the problem of upgrading weaker equivalences-such as orbit equivalence or topological conjugacy-to full smooth conjugacy when standard obstructions (e.g., periodic orbit constraints or Lyapunov exponent mismatches) vanish. The PI will develop new analytic and geometric tools to construct the desired smooth coordinate changes. 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.