A Comprehensive Program in Modern Dynamics with Emphasis on Rigidity

The Proposed program aims at advancing of several directions of research across the principal areas of the modern structural theory of dynamical systems: hyperbolic (mostly and non-uniform), partially hyperbolic, parabolic and elliptic with a particular emphasis on various kinds of rigidity phenomena and on applicability of several powerful methods across areas. New methods and insights have been introduced by the PI and his collaborators over the past years. The most recent advances allowed to obtain arithmetic structure almost everywhere smooth in the sense of Whitney on a set of arbitrarily large measure for a class of maximal rank actions (the rank of the acting group is one less than the dimension of phase space) under a very general assumption of sufficient complexity. A dichotomy was established: for such actions: either there is not exponential complexity at all, or its measure ( a version of entropy) is uniformly bounded from below by a constant that grows to infinity with dimension. Directions in the proposed program include extension of the arithmeticity program to broader classes of actions with the rank not related to dimension, applications of measure rigidity to Zimmer program, further development of the entropy theory and entropy rigidity for higher rank abelian actions, completing the program of differentiable rigidity of partially hyperbolic algebraic actions of higher rank abelian groups, the study of rigidity phenomena for totally non- hyperbolic, unipotent actions that present a striking contrast for the classical case, non-standard KAM-types invariant curve theorem and further investigation of the smooth realization problem.

Dynamical systems serve as mathematical models of time evolution of various process across the areas of natural and social sciences. It also has a surprisingly broad range of applications within core mathematical disciplines, most particularly to various areas of geometry and number theory. Within many of these contexts the 'time' is not necessarily the usual one-dimensional time but it can be multi-dimensional or, of even more general nature that is described by the key mathematical concept of group. There is a crucial difference between the classical case of one-dimensional time and that of multi-dimensional time: while in the former case various complexity and chaotic phenomena may appear gradually and chaotic behavior can and usually does coexists with ordered one within the same systems (e.g. ordered planetary motions vs. chaotic behavior of some asteroids and smaller objects in the solar system), in the latter as was established by the PI and his collaborators) complexity is often global and in fact chaos appears in a highly structured way and with high albeit calculable levels of complexity as measured by a proper version of the fundamental notion of entropy.