Differentiability of Mather’s average action and integrability on closed surfaces

In the study of Tonelli Lagrangian and Hamiltonian systems, a central role in understanding the dynamical and topological properties of the action-minimizing sets (also called Mather and Aubry sets), is played by the so-called Mather’s average action (sometimes referred to as β-function or effective Lagrangian), with particular attention to its differentiability and nondifferentiability properties. Roughly speaking, this is a convex superlinear function on the first homology group of the base manifold, which represents the minimal action of invariant probability measures within a prescribed homology class, or rotation vector (see Equation (1) for a more precise definition). Understanding whether or not this function is differentiable, or even smoother, and what are the implications of its regularity to the dynamics of the system is a formidable problem, which is still far from being completely understood. Examples of Lagrangians admitting a smooth β-function are easy to construct. Trivially, if the base manifold M is such that dimH1(M ;R) = 0 then β is a function defined on a single point and it is therefore smooth. Furthermore, if dimH1(M ;R) = 1 then a result by M. Dias Carneiro [8] allows one to conclude that β is differentiable everywhere, except possibly at the origin. As soon as dimH1(M ;R) ≥ 2 the situation becomes definitely less clear and the smoothness of β becomes a more “untypical” phenomenon. Nevertheless, it is still possible to find some interesting examples in which it is smooth. For instance, let H : TT −→ R be a completely integrable (in the sense of Liouville) Tonelli Hamiltonian system, given by H(x, p) = h(p), and consider the associated Lagrangian L(x, v) = l(v) on TT. It is easy to check that in this case, up to identifying H1(T ;R) with R, one has β(h) = l(h) and therefore β is as smooth as the Lagrangian and the Hamiltonian are. One can weaken the assumption on the completely integrability of the system and consider C-integrable systems, i.e., Hamiltonian systems that admit a foliation of the phase space by disjoint invariant continuous Lagrangian graphs, one for each possible cohomology class (see Definition 1). It is then possible to prove that in this case the associated β function is C (see Lemma 2). These observations arise the following question: with the exception of the mentioned trivial cases, does the regularity of β imply the integrability of the system?