Through the perspective of ergodic theory, if one wants to study discrete dynamical system on a nice space, it is quite desirable to look for invariant measures and study ergodic/mixing properties. However, this often does not work even for nicest group actions on nice topological spaces. Our motivating example will be the action of a non-elementary Fuchsian group on the hyperbolic plane which induces the action on S^1. Such actions generally do not admit measures which are invariant with respect to the entire group.

However, given a measure mu on a Fuchsian group, one might relax the invariance and ask whether a measure on S^1 is invariant "on average" with respect to mu. Such measures are called mu-stationary, and while one can show the existence and uniqueness of the stationary measure for very general mu, there is still no satisfactory classification which tells us when the stationary measure is singular/absolutely continuous wrt the Lebesgue measure on S^1. We will discuss the classical results and the latest advances related to this problem.