Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map φ¯F is a composition of an exact monotone twist map φ¯ with a generating function H and a vertical translation VF with VF((x,y))=(x,y−F). We show in this paper that for each ω∈R, there exists a critical value Fd(ω)≥0 depending on H and ω such that for 0≤F≤Fd(ω), the non-exact twist map φ¯F has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff (p,q)-periodic orbits for rational ω=p/q. Like the Aubry-Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value F=Fd(ω), the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.