Colloquium

Importance of invariant measures in the theory of dynamical systems and its manifold applications is widely known. One should keep in mind though that measures are linear functionals defined on the spaces of continuous functions, and natural classes of observables that appear in various branches of dynamics and its applications are usually smaller. Linear functionals defined on relevant spaces of observables are distributions and invariant distributions play the central role in wide variety of questions from calculating the rates of deviation of ergodic averages and correlation decay to classification of time changes and local classification of dynamical systems up to differentiable conjugacy.

General theory of invariant distributions for classical dynamical systems (diffeomorphisms and flows on compact manifolds) does not exist but a variety of useful methods have been developed to construct and classify invariant distributions for various classes of dynamical systems with all four principal types of infinitesimal asymptotic behavior: elliptic, parabolic, hyperbolic and partially hyperbolic.

In this talk I will give a partial overview of this complicated and diverse area, describe some recent results and mention natural but challenging open problems.