Applications of Forcing Theory to Homeomorphisms of the Closed Annulus

TL;DR

This paper applies forcing theory to surface homeomorphisms of the closed annulus, solving a conjecture on rotation sets, analyzing boundary behavior, and extending Aubry-Mather theory to area-preserving homeomorphisms.

Contribution

It introduces a forcing theory approach to analyze rotation sets of annular homeomorphisms, proving new results on rotation numbers and invariant sets.

Findings

Proves the strong form of Boyland's Conjecture for the closed annulus.

Shows bounded deviations from rotation sets for boundary-including homeomorphisms.

Establishes existence of Aubry-Mather sets for all rotation numbers in the rotation set of area-preserving homeomorphisms.

Abstract

This paper studies homeomorphisms of the closed annulus that are isotopic to the identity from the viewpoint of rotation theory, using a newly developed forcing theory for surface homeomorphisms. Our first result is a solution to the so called strong form of Boyland's Conjecture on the closed annulus: Assume $f$ is a homeomorphism of $\overline{\mathbb{A}}:=(\mathbb{R}/\mathbb{Z})\times [0,1]$ which is isotopic to the identity and preserves a Borel probability measure $\mu$ with full support. We prove that if the rotation set of $f$ is a non-trivial segment, then the rotation number of the measure $\mu$ cannot be an endpoint of this segment. We also study the case of homeomorphisms such that $\mathbb{A}=(\mathbb{R}/\mathbb{Z})\times (0,1)$ is a region of instability of $f$. We show that, if the rotation numbers of the restriction of $f$ to the boundary components lies in the interior of the rotation set of $f$, then $f$ has uniformly bounded deviations from its rotation set. Finally, by combining this last result and recent work on realization of rotation vectors for annular continua, we obtain that if $f$ is any area-preserving homeomorphism of $\overline{\mathbb{A}}$ isotopic to the identity, then for every real number $\rho$ in the rotation set of $f$, there exists an associated Aubry-Mather set, that is, a compact $f$-invariant set such that every point in this set has a rotation number equal to $\rho$. This extends a result by P. Le Calvez previously known only for diffeomorphisms.