Mathers regions of instability for annulus diffeomorphisms

Abstract

Let $f$ be a $C^{1+\epsilon }$ diffeomorphism of the closed annulus $A$ that preserves orientation and the boundary components, and $\tilde{f}$ be a lift of $f$ to its universal covering space. Assume that $A$ is a Birkhoff region of instability for $f$, and the rotation set of $\tilde{f}$ is a non-degenerate interval. Then there exists an open $f$-invariant annulus $A^{\ast }$ whose boundary intersects both boundary components of of $A$, and points $z^{+}$ and $z^{-}$ in $A^{\ast }$, such that the positive (resp. negative) orbit of $z^{+}$ converges to a set contained in the upper (resp. lower) boundary component of $A^{\ast }$ and the positive (resp. negative) orbit of $z^{-}$ converges to a set contained in the lower (resp. upper) boundary component of $A^{\ast }$. This extends a celebrated result originally proved by Mather for area-preserving twist diffeomorphisms.

17 pages, to appear in Bulletin of the London Math. Soc

References in corpus (2)

• Forcing theory for transverse trajectories of surface homeomorphisms
• Accessible points rotate as prime ends in backward or forward time