Mathers regions of instability for annulus diffeomorphisms

TL;DR

This paper extends Mather's result on regions of instability in annulus diffeomorphisms to a broader class, showing the existence of invariant annuli with boundary points exhibiting specific orbit convergence behaviors.

Contribution

It generalizes Mather's theorem to non-area-preserving diffeomorphisms with non-degenerate rotation sets, identifying invariant annuli with boundary orbit convergence.

Findings

Existence of an invariant annulus with boundary points converging to boundary components.

Extension of Mather's theorem to non-area-preserving cases.

Identification of boundary orbit behaviors in the invariant annulus.

Abstract

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