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

作者：Qin, Wen-Xin^*; Wang, Ya-Nan

来源：Journal of Differential Equations, 2018, 264(11): 7006-7021.

DOI：10.1016/j.jde.2018.02.007

• Aubry-Mather theory
• Non-exact twist map
• Denjoy minimal set
• Birkhoff periodic orbit
• Depinning force

A non-exact monotone twist map (phi) over bar (F), is a composition of an exact monotone twist map (phi) over bar with a generating function H and a vertical translation V-F with V-F((x, y)) = (x, y - F). We show in this paper that for each omega is an element of R, there exists a critical value F-d (omega) >= 0 depending on H and omega such that for 0 <= F <= F-d (omega), the non-exact twist map (phi) over bar (F), has an invariant Denjoy minimal set with irrational rotation number omega lying on a Lipschitz graph, or Birkhoff (p, q)-periodic orbits for rational omega = 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 = F-d(omega), the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.

• 出版日期2018-6-5
• 单位苏州大学; 南京师范大学