Aubry-Mather theory for contact Hamiltonian systems III

By exploiting the contact Hamiltonian dynamics (T* M × ℝ, Φ_t) around the Aubry set of contact Hamiltonian systems, we provide a relation among the Mather set, the Φ_t-recurrent set, the strongly static set, the Aubry set, the Mañé set, and the Φ_t-non-wandering set. Moreover, we consider the strongly static set, as a new flow-invariant set between the Mather set and the Aubry set in the strictly increasing case. We show that this set plays an essential role in the representation of certain minimal forward weak Kolmogorov-Arnold-Moser (KAM) solutions and the existence of transitive orbits around the Aubry set.