  A wandering domain for a diffeomorphism is an open connected set whose
iterates are pairwise disjoint. We endow A^n = T^n x R^n with its usual exact
symplectic structure. An integrable diffeomorphism {\Phi}^h, i.e. the time-one
map of a Hamiltonian h which depends only on the action variables, has no
nonempty wandering domains. The aim of this paper is to estimate the size
(measure and Gromov capacity) of wandering domains in the case of an exact
symplectic perturbation of {\Phi}^h , in the analytic or Gevrey category. Upper
estimates are related to Nekhoroshev theory, lower estimates are related to
examples of Arnold diffusion. This is a contribution to the "quantitative
Hamiltonian perturbation theory" initiated in previous works on the optimality
of long term stability estimates and diffusion times; our emphasis here is on
discrete systems because this is the natural setting to study wandering
domains.
