Applications of forcing theory to homeomorphisms of the closed annulus

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.