Abstract: Since several years it is known that certain discretizations
for the geodesic flow on hyperbolic surfaces of emph{finite area} allow
to provide a dynamical characterizations of Maass cusp forms and a
dynamical construction of their period functions. An important
ingredient for these results is the characterization of Laplace
eigenfunctions in parabolic cohomology by Bruggeman--Lewis--Zagier.

We discuss an extension of these results to Hecke triangle surfaces of
emph{infinite area} and automorphic forms that are more general than
Maass cusp forms. This is joint work with R. Bruggeman.