Topics in Dynamical Systems, Ergodic Theory and Geometry

The project covers a classification of Anosov (normally hyperbolic) and certain classes of partially hyperbolic actions of higher-rank commutative groups, both discrete and continuous, on compact manifolds, classification of several classes of smooth actions of lattices in higher-rank semi-simple Lie groups, a systematic study of invariant measures for Anosov and partially hyperbolic algebraic actions of discrete and continuous higher-rank abelian groups, development of a general theory of hyperbolic measures for actions of higher-rank abelian groups, and developments of a uniform theory of invariant distributions for various classes of dynamical systems. Among the principal tools are smooth ergodic theory (Lyapunov characteristic exponents and non-uniform hyperbolicity), the theory of non-stationary normal forms, geometric super-rigidity and the theory of group representations. The theory of dynamical systems is the mathematical foundation of the rapidly developing fields of non-linear dynamics and chaos theory which provides these fields with their principal paradigms and tools of rigorous analysis. Those paradigms in turn play a key role in the development and analysis of mathematical models for numerous problems within natural and social sciences and engineering. The standard setup in dynamics considers time as one-dimensional, either discrete or continuous. One of the central conclusions is that in a variety of situations, the orbit structure is rich and its robust features can be described by well-understood symbolic models. The research under the present grant looks at the situation when the time is multi-dimensional while the phase space is still finite-dinemsional, i.e. the state of a system under consideration can be described by a finite set of numerical parameters. The leading paradigms in these cases turn out to be strikingly different. On the one hand, the symbolic models are no longer valid. On the other, the orbit structure turn out to be 'rigid', i.e. not only its robust f eatures, but much more subtle structure, including the fine statistics of asymptotic behavior, do not change under perturbations. The project intends to identify and systematically study a set of 'universal models' which replace the symbolic models in the case of one-dimensional time.