Towards a Classification for quasiperdiodically forced circle homeomorphisms

Poincaré's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms homotopic to the identity, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempted classification, and in fact we define a system as $\rho$-bounded if such deviations are bounded and as $\rho$-unbounded otherwise. For the $\rho$-bounded case we prove a close analogue of Poincaré's result: if the rotation number is rationally related to the rotation rate on the base then there exists an invariant strip (the appropriate analogue for fixed or periodic points in this context), otherwise the system is semi-conjugate to an irrational translation of the torus. In the $\rho$-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive