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 to the problem in arithmetic quantum chaos. There are several major directions in the proposed program: 1. Local differentiable rigidity for partially hyperbolic actions of higher rank abelian groups with the emphasis on the combination of the dynamical systems, harmonic analysis/group representation and geometric methods. 2. Global rigidity of Anosov actions, using various approaches based on invariant rigid geometric structures. 3. Rigidity of invariant measures using the innovative high entropy and low entropy methods in the positive entropy case as well as new approaches to the zero entropy case. 4. Problems of quantum unique ergodicity and existence of scars for Finsler geodesic flows and billiards in polygons. 5. Precise asymptotic and multiplicative lower bounds for the growth of the number of periodic orbits for broad classes of dynamical systems with non-uniformly hyperbolic behavior. 6. The problem of smooth realization of measurable dynamical systems.
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. Among the latter are: 1) problems of simultaneous approximation of several irrational numbers by rationals and 2) connection between the behavior of certain class of quantum mechanical systems and their classical limits when Plank constant goes to zero. The central idea is that certain properties of classical limits (such as hyperbolicity or ``chaos" on the one hand and presence of certain types of periodic orbits on the other) is reflected in the behavior of quantum systems such as ``unifrom distibution of quantum states " and ``scars".
