This project aims to make progress in several directions of research across the principal areas of the modern structural theory of dynamical systems: hyperbolic (uniform and nonuniform), partially hyperbolic, parabolic, and elliptic. It will put particular emphasis on various kinds of rigidity phenomena and on applicability of several powerful methods across those areas. New methods and insights have been introduced by the principal investigator and his collaborators over the past years that have led to significant advances in the understanding of rigidity of invariant measures and the differentiable rigidity of orbit structure for actions of higher rank Abelian groups. The progress achieved based on these methods has engendered fruitful applications to Diophantine approximation problems in number theory and provided the first examples of the existence of invariant geometric structure for large classes of actions. Avenues of pursuit in the project include the following: tame and wild behavior in the classification of Anosov systems up to a differentiable conjugacy, global rigidity of hyperbolic measures for actions of higher rank Abelian groups and applications to the Zimmer program, completing the program of differentiable rigidity of partially hyperbolic algebraic actions of higher rank Abelian groups, applications of the theory of unitary group representations and the KAM method to rigidity of unipotent homogeneous actions, commencement of a comprehensive program of investigation of nonhomogeneous parabolic systems, nonstandard KAM-type invariant curve theorems, and low-dimensional systems with zero topological entropy.
Dynamical systems serve as mathematical models for the time-evolution of processes that range across the spectrum of the natural and social sciences. They also have a surprisingly broad range of applications in core mathematical disciplines, most particularly in various areas of geometry and number theory. Within many of these contexts the term "time" does not necessarily connote the usual one-dimensional time but can be multidimensional or of an even more general nature that is captured by the key mathematical concept of a "group." The crucial difficulty that impedes efforts in obtain a comprehensive understanding of important models can be described as follows: while it is often relatively easy to establish existence of some initial conditions that produce chaotic behavior, proving that chaotic behavior exists in most or many dynamical systems (in the sense of so-called volume in the phase-space) is beyond the reach of present or even anticipated mathematical methods. The principal investigator and his collaborators have discovered that, for systems with multidimensional time and under certain very general conditions, this difficulty can be overcome: global conditions of a topological or dynamical nature that at first glance would seem to guarantee only the existence of some chaotic orbits actually imply the existence of a set of such orbits that fill a positive volume in phase-space or, in certain cases, even provide a complete description of the orbit structure. This seemingly technical fact could have important implications for physics and engineering.