**Parabolic dynamics and Anisotropic Banach spaces
**

We investigate the relation between the distributions appearing in the study of ergodic averages of parabolic ows (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 ow on the torus. Our main result is that, roughly, the growth of ergodic averages of a parabolic ows 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. Joint work with C. Liverani.