Construction of a Nekhoroshev like result for the asteroid belt dynamical system

We investigate the possibility of obtaining a Nekhoroshev like result for the dynamical system describing the motion of an asteroid in the main belt. From the mathematical point of view this is a new result since the problem is degenerate and we want to control also the motion of degenerate actions. We find that there are regions, such as the resonances of low order among the fast angles (mean motion resonances), where a Nekhoroshev like result cannot be proved a priori. Conversely, we are able to confine the motions in the mean motion resonances of logarithmically large order in the perturbation parameters, as well as in the non-resonant region. We discuss also the connection with the existence of invariant tori.