  We investigate the relation between the distributions appearing in the study
of ergodic averages of parabolic flows (e.g. in the work of Flaminio-Forni) and
the ones appearing in the study of the statistical properties of hyperbolic
dynamical systems (i.e. the eigendistributions of the transfer operator). In
order to avoid, as much as possible, technical issues that would cloud the
basic idea, we limit ourselves to a simple flow on the torus. Our main result
is that, roughly, the growth of ergodic averages of a parabolic flows is
controlled by the eigenvalues of a suitable transfer operator associated to the
renormalising dynamics. The conceptual connection that we illustrate is
expected to hold in considerable generality.
