The topic of the proposed meeting is at the crossroad between hyperbolic and parabolic classical dynamics and quantum dynamics (thru microlocal analysis). We believe that each of these three fields has both problems and solutions that can be used to substantially advance the others. Unfortunately, there
exists a considerable language barrier that makes very difficult the communication.

Hence the idea is to bring together some of the top people in each field, plus some younger researchers, in an informal environment in which each group can try to communicate the basic facts to the others in order to develop a common Language and background.

Please see the section **Planning activities** for more detailed information about the structure of the meeting.

In recent years methods from analysis, in particular Fourier analysis and microlocal analysis, applied to the study of distributional invariants of dynamical
systems and more generally to the action of dynamical systems on distribution
spaces, has gained a considerable prominence in the field of Dynamical Systems (see **(2, 8, 9, 1, 22)**, just to mention a few).

The study of the evolution of distributions (rather than points, or measures) has proved to be a powerful tool in the study of the statistical properties of hyperbolic systems **(3, 20, 19, 2, 17)** and of cohomological equations and speed of ergodicity for parabolic systems (see for instance **(15, 16, 13, 14, 4, 5, 12, 21, 6)**. We remark that the study of parabolic dynamical systems is often based on the analysis of a hyperbolic (or partially hyperbolic) renormalization dynamics (basic examples include translation flows, horocycle
flows and nilflows). In addition, in hyperbolic dynamics, in particular in the study of Anosov flows, methods from microlocal analysis have led to a unified approach to the study of the classical and quantum evolution **(10, 11)**.

Given the above mentioned developments, it appears that there exists a very
close connection between methods and ideas in several fields, in particular the increasing relevance of distributional spaces and distributional invariants of classical and quantum evolution. Some efforts to elucidate such a connection already exists **(13, 7, 18)** and point toward the need for further study. For instance it has been understood that whenever a parabolic system is renormalized by a Anosov map or (contact) flow, then invariant distributions for the parabolic dynamics (which yield obstructions to solving cohomological equations) appear as distributional eigenvectors for an appropriate transfer operator of the Anosov system. Natural questions which need to be addressed are the generalization of analytical methods to more general Anosov systems, to partially hyperbolic systems and to renormalization dynamics on "moduli" spaces of parabolic systems. Finally, analytical methods need to be developed to investigate parabolic systems which are non-renormalizable.