Connecting orbits of time dependent Lagrangian systems

We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one proposed by Mather in [7]. However, its advantage is that it contains most of the results of Birkhoff and Mather on twist maps. A very natural class of problems in dynamical systems is the existence of orbits connecting prescribed regions of phase space. There are several important open questions in this line, like the one posed by Arnold : Is a generic Hamiltonian system transitive on its energy shells? Birkhoff’s theory of regions of instability of twists maps, recently extended by Mather using variational methods and by Le Calvez, provide very relevant results in that direction. In short, these works establish the existence, for a certain class of mappings of the annulus, of orbits visiting in turn prescribed regions of the annulus under the hypothesis that these regions are not separated by a rotational invariant circle. John Mather has opened the way to a generalization in higher dimension of this celebrated theory by proposing what seems to be the appropriate setting i.e. time dependent positive definite Lagrangian systems. In this setting, he has obtained the existence of families of invariant sets generalizing the well known Aubry-Mather invariant sets of twist maps. Then he stated in 1993 a result on the existence of orbits visiting in turn neighborhoods of an arbitrary sequence of these invariant sets. However, the work of Mather is not a complete achievement since there are no relevant example in high dimension to which it can be applied, and since it is not completely optimal even in the case of Twist maps. There are examples where two Aubry-Mather sets of a twist map are not separated by a rotational invariant circle, hence can be connected by an orbit, but where this can’t be seen by the result of Mather. In the present paper, we state a new result on the existence of connecting orbits in higher dimension, with a full self-contained proof. This result is very close to the one of Mather, and the main ideas of the proof are the ones he introduced. Our result has the advantage that it is optimal when applied to the twist map case, but it does not contain the result of Mather, which we were not able to prove. 1 It is still an open question whether these results may be applied to interesting example in higher dimension 2. On one hand, it is encouraging that this result is optimal when restricted As it is written in [9], the proof contains a gap which I am not able to fill. 2 Just before I finished this text, John Mather has announced that he had been able to prove a great result on Arnold diffusion, so the full achievement of the method may soon be reached.