  These notes are based on lectures given by the author at the Summer School on
Teichm\"uller dynamics, mapping class groups and applications in Grenoble,
France, in June 2018 and at the Oberwolfach Seminar on Anisotropic Spaces and
their Applications to Hyperbolic and Parabolic Systems in June 2019. We derive
results about the so-called Ruelle resonances and the asymptotics of
correlations for several classes of systems from known results on cohomological
equations and invariant distributions for the respective unstable vector
fields. In particular, we consider pseudo-Anosov diffeomorphisms on surfaces of
higher genus, for horocycle flows on surfaces of constant negative curvature
and for partially hyperbolic automorphisms of Heisenberg 3-dimensional
nilmanfolds. Ruelles resonances for pseudo-Anosov maps with applications to the
cohomological equation for their unstable translation flows was recently
studied in depth by F. Faure, S. Gou\"ezel and E. Lanneau [FGL] by methods
based on the analysis of the transfer operator of the pseudo-Anosov map. Ruelle
resonances for geodesic flows on hyperbolic compact manifolds of any dimension
and of partially hyperbolic automorphisms of Heisenberg 3-dimensional
nilmanfolds are studied by general results of Dyatlov, Faure and Guillarmou
[DFG] and Faure and Tsujii [FT15] based on methods of semi-classical analysis.
These works do not derive results on cohomological equations for unstable flows
or horospherical foliations of these systems.
