Aubry-Mather theory for contact Hamiltonian systems III

In view of Aubry-Mather theory and weak KAM theory in Hamiltonian systems, the Aubry set plays a central role in exploiting the dynamics of action minimizing orbits. Let H(x,p,u) be non-decreasing in u. If H is independent of u , it is well known that the Aubry set is chain-recurrent. Unfortunately, it is not true in the contact setting. The Aubry set may contain wandering points. In this work, we provided a criterion for the existence of a pseudo-orbit of the contact Hamiltonian system from one given point X_1 ∈ T ∗ M × R to another given point X_2 ∈ T ∗ M ×R . As a consequence, we proved the non-wandering property of the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set.