Rotation Numbers of Projectivities

Let D be a nonempty open convex region in R2, bounded by a closed curve C. For any a ∈ D we define a map ƒa : C → C by requiring that for each x ∈ C, the point ƒa(x) be the point of C other than x which lies on the line through x and a. If A ≔ (a1, a2, ..., an), where a1, a2, ..., an ∈ D, then we set ƒA ≔ ƒan ∘ ƒan−1 ∘• • •∘ ƒa1. Since it is an orientation preserving homeomorphism of C onto itself and C is homeomorphic to a circle, we can speak about the rotation number of ƒA, which we denote by ρ(A). We investigate the dynamical properties of ƒA depending on C and ρ(A). In particular, we show that if n ≥ 3 and k ≥ 1 are integers and ρ(A) = (2k + 1)/2 for some A ∈ Dn, then ƒ2A = id for all A ∈ ρ−1 ((2k + 1)/2) if and only if C is an ellipse.