Nekhoroshev stability of quasi-integrable degenerate Hamiltonian systems

A perturbation of a degenerate integrable Hamiltonian system has the form $H = h(I) + \varepsilon f(I,\varphi , p , q )$ with $(I, \varphi) \in \mathbf{R}^n \times \mathbf{T}^n, (p,q) \in \mathcal{B} \subseteq \mathbf{R}^{2 m}$ and the two-form is $dI \wedge d\varphi + dp \wedge dq$. In the case $h$ is convex, Nekhoroshev theorem provides the usual bound to the motion of the actions $I$, but only for a time which is the smaller between the usual exponentially-long time and the escape time of $p,q$ from $\mathcal{B}$. Furthermore, the theorem does not provide any estimate for the "degenerate variables" $p,q$ better than the a priori one $\dot{p},\dot{q} \sim \varepsilon$, and in the literature there are examples of systems with degenerate variables that perform large chaotic motions in short times. The problem of the motion of the degenerate variables is relevant to understand the long time stability of several systems, like the three body problem, the asteroid belt dynamical system and the fast rotations of the rigid body.
In this paper we show that if the "secular" Hamiltonian of $H$, i.e. the average of $H$ with respect to the fast angles $\varphi$, is integrable (or quasi-integrable) and if it satisfies a convexity condition, then a Nekhoroshev-like bound holds for the degenerate variables (actually for the actions of the secular integrable system) for all initial data with initial action $I(0)$ outside a small neighbourhood of the resonant manifolds of order lower than $\ln \dfrac{1}{\varepsilon}$. This paper generalizes a result proved in connection with the problem of the long-time stability in the Asteroid Main Belt [9,13].