The central theme of the proposed program is the study of various aspects of rigidity in dynamics. New methods and insights have been recently introduced by the PI and his collaborators which led to a significant progress in the problem of rigidity of invariant measures and the differentiable rigidity of orbit structure for actions of higher rank abelian groups. Advances achieved based on these methods engendered fruitful applications to Diophantine approximation problems in number theory and provided first examples of existence of invariant geometric structure for large classes of actions. There are several major directions in the proposed program:
1.Local differentiable rigidity for partially hyperbolic and parabolic actions of higher rank abelian groups with the emphasis on the combination of the dynamical systems, harmonic analysis/group representation and geometric methods.
2. Finding new methods in the theory of measure rigidity for algebraic dynamical systems.
3. Existence of invariant geometric structures for general classes of actions of higher rank abelian groups determined by global topological, homotopical or dynamical conditions.
4.Classification of Anosov systems up to differentiable conjugacy vis various classes of moduli.
5.Construction of smooth realization of Gaussian dynamical systems.
6.Application of the theory of unitary group representations to the cohomological problems in dynamics.
Mathematical concept of "rigidity" has many facets. Its simplest and most basic manifestations can be seen from the following elementary example: a small number of equations or inequalities of a special type may imply much larger number of equation. For example, if the arithmetic mean on n numbers coincides with the geometric mean (one equation) then the numbers are all equal ( n-1 equations). An example from the PI's earlier research is conceptually similar albeit technically much more sophisticated: a compact surface of negative curvature, i.e. a bounded geometric shape where any geodesic triangle has the sum of its angles less than 180 degrees, for which two numbers characterizing global and statistical volume growth (topological and metric entropy) coincide has constant negative curvature, i.e. the sum of the angles of a geodesic triangle is uniquely determined by the area. The research under the present grant involves both deeper investigation of rigidity phenomena for dynamical systems with multi-dimensional time, and expansion and development of striking application to several areas of mathematics and mathematical physics.
Here is an interpretation of some results from the area 3. above. Investigation of chaotic behavior in deterministic dynamical systems plays a central role in application of the modern theory of dynamical systems to various areas on natural and social sciences. Here is the crucial difficulty which impedes efforts in more comprehensive understanding of important models: while it is often relatively easy to establish existence of some initial conditions which produce chaotic behavior, proving chaotic behavior of most or many (in the sense of volume in the phase space) is beyond the present on even anticipated mathematical methods. PI and his collaborators discovered that for systems with multidimensional time under certain very general conditions this difficulty is overcome: global conditions of topological or dynamical nature which a priori guarantee only existence of some chaotic orbits in fact imply existence of such orbits which fill positive volume in the phase space.
