Periodic point free homeomorphisms of the open annulus: from skew products to non-fibred maps

Abstract

The aim of this paper is to study and compare the dynamics of two classes of periodic point free homeomorphisms of the open annulus, homotopic to the identity. First, we consider skew products over irrational rotations (often called quasiperiodically forced monotone maps) and derive a decomposition of the phase space that strengthens a classification given by J. Stark. There exists a sequence of invariant essential embedded open annuli on which the dynamics are either topologically transitive or wandering (from one of the boundary components to the other). The remaining regions between these annuli are densely filled by so-called invariant minimal strips, which serve as natural analogues for fixed points of one-dimensional maps in this context. Secondly, we study homeomorphisms of the open annulus which have neither periodic points nor wandering open sets. Somewhat surprisingly, there are remarkable analogies to the case of skew product transformations considered before. Invariant minimal strips can be replaced by a class of objects which we call invariant circloids, and using this concept we arrive again at a decomposition of the phase space. There exists a sequence of invariant essential embedded open annuli with transitive dynamics, and the remaining regions are densely filled by invariant circloids. In particular, the dynamics on the whole phase space are transitive if and only if there exists no invariant circloid and if and only if there exists an orbit which is unbounded both above and below.