Forcing theory for transverse trajectories of surface homeomorphisms

TL;DR

This paper introduces a new topological orbit forcing theory for surface homeomorphisms, leading to novel proofs and results in surface dynamics, including classifications, rotation set properties, and conjecture resolutions.

Contribution

It develops a novel orbit forcing theory using maximal isotopies and transverse foliations, extending existing classifications and resolving conjectures in surface dynamics.

Findings

Extended classification of zero entropy maps on $S^2$ for non-wandering homeomorphisms.

Proved the rotation set of Hamiltonian annulus homeomorphisms either singleton or contains zero.

Constructed convex sets not realizable as rotation sets of torus homeomorphisms.

Abstract

This paper studies homeomorphisms of surfaces isotopic to the identity by means of purely topological methods and Brouwer theory. The main development is a novel theory of orbit forcing using maximal isotopies and transverse foliations. This allows us to derive new proofs for some known results as well as some new applications, among which we note the following: we extend Franks and Handel's classification of zero entropy maps of $S^2$ for non-wandering homeomorphisms; we show that if $f$ is a Hamiltonian homeomorphism of the annulus, then the rotation set of $f$ is either a singleton or it contains zero in the interior, proving a conjecture posed by Boyland; we show that there exist compact convex sets of the plane that are not the rotation set of some torus homeomorphisms, proving a first case of the Franks-Misiurewicz Conjecture; we extend a bounded deviation result relative to the rotation set to the general case of torus homeomorphisms.