On Properties of the Vertical Rotation Interval for Twist Mappings II

In this paper we consider twist mappings of the torus, T : T2 → T2, and their vertical rotation intervals ρV (T ) = [ρ − V , ρ + V ], which are closed intervals such that for any ω ∈]ρV , ρ+V [ there exists a compact T -invariant set Qω with ρV (x) = ω for any x ∈ Qω , where ρV (x) is the vertical rotation number of x. In case ω is a rational number, Qω is a periodic orbit (this study began in [1] and [2]). Here we analyze how ρV and ρ + V behave as we perturb T when they assume rational values. In particular we prove that, for an interesting class of mappings, these functions are locally constant at rational values.