Towards a Classification for Quasiperiodically Forced Circle Homeomorphisms

Abstract

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 ρ-bounded if such deviations are bounded and as ρ-unbounded otherwise. For the ρ-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 ρ-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive.

Denjoy constructions for fibered homeomorphisms of the torus

“…Up to a linear fibered conjugacy, we can assume that We choose a lift F of f to T 1 ×R, and consider the liftK of K. By the choice of I, for every z ∈K ∩ (I × R) there exists τ (z) ∈ 1 2 Z such that z ∈ I × (τ (z), τ (z) + 1/4). Now let θ, n be such that θ and θ + nω belong to I ∩ Θ 2 .…”

“…The interest in transitive but non-minimal dynamics in this kind of map is motivated by a recent classification result in [1], which we briefly discuss in order to motivate the problem. ρ(F ) = lim n→∞ (F n θ (x) − x)/n exists and does not depend on (θ, x) ∈ T 1 × R. Furthermore, the convergence in (1.2) is uniform [5].…”

“…However, unlike the one-dimensional case, the deviations from the average rotation, given by (1.3) D n (θ, x) := F n θ (x) − x − nρ(F ) , need not be uniformly bounded in n, θ, x anymore. 1 This gives rise to a basic dichotomy: a homeomorphism f ∈ F is called ρ-bounded if sup n,θ,x |D n (θ, x)| < ∞ and ρ-unbounded otherwise.…”

“…[7]) holds, with invariant strips playing the role of periodic orbits in the unforced case: Theorem 1.1 (Theorems 3.1 and 4.1 in [1]). …”

“…To get the reverse inclusion we have to make a more careful choice of the sequence (ε n ). For a fixed n ≥ k, denote by O n,k the set of continuous graphs ∆ such that p 1 …”