Remarks on skew products over irrational rotations

We prove several results of the orbit behavior of skew product diffeomorphisms generated by quasi-periodic differential systems. The first diffeomorphism is derived from a periodic differential equation on the circle by means of a construction proposed by Z. Opial to get a scalar quasi-periodic equation with all its solutions bounded but without in almost periodic solution. We consider both possible cases for the irrational rotation number, transitive and singular (intransitive). The main result for a, transitive case is that the related skew product diffeomorphism has a foliation into invariant curves with pure irrational rotation on each curve (being the same for each curve). For intransitive case, we get invariant sets of two types: a collection of continuous invariant curves and invariant sets being dimensionally inhomogeneous ones. Section 3 is devoted to perturbations of a skew product diffeomorphism over an irrational rotation being initially foliated into invariant curves. We prove an analog of Poincare-Pontryagin theorem which sets conditions when a perturbation of a. one-degree-of-freedom Hamiltonian system (given in an annulus and written down in action-angle variables) has limit cycles. Our theorem provides sufficient conditions when a perturbation of a foliated skew product diffeomorphism has isolated invariant curves (asymptotically stable or unstable). In Sec. 4 we present some results on the geometry of skew product diffeomorphisms derived by a quasi-periodic Riccati equation.