Rotation sets for orbits of degree one circle maps

Let F be the lifting of a circle map of degree one. In [Bamón et al., 1984] a notion of F-rotation interval of a point x ∈ S1 was given. In this paper we define and study a new notion of a rotation set of a point which preserves more of the dynamical information contained in the sequences {Fn(y)}n=0∞ than the one preserved from [Bamón et al., 1984]. In particular, we characterize dynamically the endpoints of these sets and we obtain an analogous version of the Main Theorem of [Bamón et al., 1984] in our settings.