Stefano MARMI

DIPARTIMENTO DI MATEMATICA Ulisse Dini
VIALE MORGAGNI 67/A
50134 FIRENZE
ITALY
MARMI@udini.math.UNIFI.IT

Materialization of resonances in the linear homological equation and in the semistandard map

Abstract: The problem of the materialization of resonances in perturbation series of KAM theory has been first addressed by V. I. Arnol'd in his study of the diffeomorphisms of the circle. In this case the analytical and the dynamical origin of the divergence of the series is quite weel understood thanks to the fundamental contributions of Herman, Yoccoz and Perez-Marco. Much less is known is more general situations including Hamiltonian systems. In the first part of this lecture I will briefly describe a result on the quasianalytic monogenic properties of the solutions of one dimensional linear homological equations obtained in collaboration with D. Sauzin. Our results give a partial answer to a question of M. Herman. In the second part I will describe a result on the behaviour at resonances of the linearization of the semi--standard map and state some open problems on the Borel summability and quasianalytic properties of the linearization.This second part is joint work with A. Berretti and D. Sauzin.