Invariant circles and rotation bands in monotone twist maps

Abstract

In this paper we show that the existence of certain orbits or minimal sets in an area-preserving monotone twist map is necessary and sufficient for the non-existence of invariant circles with specified rotation numbers. The necessity of these conditions follows from classic results of Birkhoff and recent results of Mather. The sufficiency of these conditions depends on the notion of a rotation band which associates a set of rotation numbers with a given orbit or invariant set. We also make some remarks on Mather's paper [M4]. In particular, we use his main theorem to give a lower bound on the width of the interval of rotation numbers associated with the zone of instability that “contains” the irrational ω whenf has no invariant circle with rotation number ω.