  We introduce two numerical conjugacy invariants for dynamical systems -- the
complexity and weak complexity indices -- which are well-suited for the study
of "completely integrable" Hamiltonian systems. These invariants can be seen as
"slow entropies", they describe the polynomial growth rate of the number of
balls (for the usual "dynamical" distances) of coverings of the ambient space.
We then define a new class of integrable systems, which we call decomposable
systems, for which one can prove that the weak complexity index is smaller than
the number of degrees of freedom. Hamiltonian systems integrable by means of
non-degenerate integrals (in Eliasson-Williamson sense), subjected to natural
additional assumptions, are the main examples of decomposable systems. We
finally give explicit examples of computation of the complexity index, for
Morse Hamiltonian systems on surfaces and for two-dimensional gradient systems.
