Parabolic dynamics via anisotropic spaces

I will present a strategy which allows to study ergodic averages and cohomological equations related to parabolic dynamics by means of hyperbolic renormalizations, exploiting transfer operator techniques on anisotropic Banach spaces.

After reporting on some cases where this strategy has been successfully applied, I will present recent work both on the horocycle flow and on parabolic flows on surfaces with singularities.

Some results are joint work with C. Liverani and some with M. Artigiani.