Small Divisors, Stability and Rigidity in Smooth Dynamics Systems

The mathematical theory of dynamical systems aims to describe the behavior of systems that evolve with time, such as the motion of the planets or of gas particles in the air. The modern theory has accumulated a corpus of concepts and tools that are fundamental in understanding classical mechanics and other problems from mathematical physics. One of its central topics is the stability (and instability) study of conservative systems. The principal investigator (PI) intends to push this study from various points of view including topological stability, measure-theoretical or statistical stability, effective or long finite time stability, as well as rigidity under perturbations. This project will also provide opportunities for graduate student research training. One of the main objectives of the project is to push the stability and rigidity results of KAM theory (after Kolmogorov Arnold and Moser) in various new directions, namely when one or several hypotheses of the classical KAM theory are not verified such as absence of any transversality condition, lack of regularity, combination with hyperbolic or parabolic dynamics, global (non-perturbative) results, presence of noise or presence of dissipative terms, large or infinite dimension. The project also encloses several themes related to diffusion and instability in conservative dynamics. Resonances and small divisors problems are central to the stability study and provide a unifying aspect in the following topics, to be investigated during the project's span: • Transfer of energy in Hamiltonian systems with excitation and partial damping. • Lyapunov instability of elliptic equilibria for real analytic Hamiltonians. • KAM stability without any transversality condition. • Stability in non-smooth Hamiltonian systems. • KAM for dissipative systems. • Stability in Hamiltonian systems in large and infinite number of degrees of freedom. • Linearization beyond the Diophantine condition. • Local rigidity of affine higher rank actions on the torus. 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.