Geometric Proofs of Mather’s Connecting and Accelerating Theorems

In this paper we present simplified proofs of two important theorems of J.Mather. The first (connecting) theorem [Ma2] is about wandering trajectories of exact area-preserving twist maps naturally arising for Hamiltonian systems with 2 degrees of freedom. The second (accelerating) theorem is about dynamics of generic time-periodic Hamiltonian systems on twotorus (2.5-degrees of freedom). Mather [Ma6] proves that for a generic timedependent mechanical Hamiltonian there are trajectories whose speed goes to infinity as time goes to infinity, in contrast to time-independent case, where there is a conservation of energy. The results of this paper are not new and the main purpose is to present simplified geometric proofs of two important theorems of J. Mather [Ma2, Ma6]. Both theorems are particular examples of instabilities in Hamiltonian systems or what is sometimes called Arnold’s diffusion. Recently Mather [Ma7] annonced a proof of existence of Arnold’s diffusion for a generic nearly integrable Hamiltonian systems with 2.5 and 3-degrees of freedom using his variational approach developed in [Ma2]–[Ma6]. The first (connecting) Mather’s theorem says that inside of a Birkhoff region of instability there are trajectories connecting any two Aubry-Mather sets, i.e. given any two Aubry-Mather sets Σω and Σω′ inside of a Birkhoff region of instability there is a trajectory α-asymptotic to Σω and ω-asymptotic to Σω′ . Recently J. Xia [X] gave a simplified proof of the first result using the same variational approach as Mather. The second (accelerating) theorem says that a “generic” Hamiltonian time periodic system on the 2-torus T has trajectories whose speed goes to infinity as time goes to infinity. Different from [Ma6] proofs of this result are given by Bolotin-Treschev [BT] and Delshams-de la Llave-Seara [DLS1]. Our proof of the second theorem combines a geometric approach and Mather’s variational approach. The second theorem is proved using ideas from the proof of the first theorem. Let’s give the rigorous statement of both results. First show that exact area-preserving twist maps naturally arise for Hamiltonian systems with two degrees of freedom. Indeed, let H : R → R be a C-smooth function and consider the corresponding Hamiltonian system The author is partially supported by AIM fellowship.