A comprehensive program in modern dynamics: flexibility, rigidity and low complexity systems

The proposed program will advance research across the principal areas of the modern structural theory of dynamical systems. A classical dynamical system is a space together with a map or flow of the space into itself. Sometimes there is also a group of symmetries of the space. If the group of symmetries is complicated enough, the system is often 'rigid'. This means the important dynamical properties of the symmetries cannot be changed without destroying the entire system. In contrast, the classical systems, with a single map or flow, are often quite flexible. This research program is aimed at understanding this flexibility.

In hyperbolic dynamics there is a contrast between behavior of classical smooth systems (diffeomorphisms and flows) and actions of higher rank abelian groups. The latter, which have been at the center of the PI's research during the last decade, exhibit remarkable rigidity of behavior. Classical systems, on the other hand, are quite flexible. The principal challenge of the flexibility program is that numerical dynamical invariants can only be precisely calculated in very few cases, mostly of algebraic origin. Most known constructions are perturbative and hence at best would allow to cover a small neighborhood of the values allowed by the model, or more often, not even that, since homogeneous systems are often 'extremal'. So establishing flexibility calls for non-perturbative or large perturbation constructions in large families to cover possible values of invariants. This calls for a combination of methods from the theory of Lyapunov characteristic exponents, smooth ergodic theory, and geometry. While the principal problems are relatively easy to explain to a fairly broad audience of mathematicians and scientists familiar with the key notions of the modern theory of dynamical systems, the point of view is quite new and has been explicitly formulated by the PI within the last few years. A second direction of research is the further development of the rigidity program for actions of higher rank abelian groups. The combinations of techniques and insights that goes under the name 'non-uniform measure rigidity' resulted in the almost definitive description of maximal rank actions. Those actions turn out to have an arithmetic nature from measure-theoretic and, with proper qualifications, also geometric point of view. There are non-trivial implications for the topology of manifolds that can carry such actions. The proposed research includes further study of the topology of maximal rank actions, as well as an extension of arithmeticity results to broader classes of actions whose rank is not related to the dimension of the ambient manifold. Additional directions of research deal with zero entropy systems and include several areas of study in both parabolic (polynomial complexity) and elliptic (low complexity) systems.