Mather's regions of instability for annulus diffeomorphisms

Abstract

AbstractLet be a diffeomorphism of the closed annulus that preserves orientation and the boundary components, and be a lift of to its universal covering space. Assume that is a Birkhoff region of instability for , and the rotation set of is a nondegenerate interval. Then there exists an open ‐invariant essential annulus whose frontier intersects both boundary components of , and points and in , such that the positive (resp., negative) orbit of converges to a set contained in the upper (resp., lower) boundary component of and the positive (resp., negative) orbit of converges to a set contained in the lower (resp., upper) boundary component of . This extends a celebrated result originally proved by Mather in the context of area‐preserving twist diffeomorphisms.