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.
