and

Let fθ(x) = F0(x) + θ (mod1), x ∈ S1, θ ∈ [0, 1] be a family of preserving orientation circle homeomorphisms with a single break point xb, i.e. with a jump in the first derivative F0 at the point x = xb. Suppose that F ′0(x) is absolutely continuous on [xb, xb+1] and F ′′0 (x) ∈ Lα([0, 1]) for some α> 1. Consider fθ with rational rotation number ρθ = p q of rank n, i.e. p q = [k1, k2,..., kn]. We prove that for sufficiently large n, the homeomorphism fθ has either a unique periodic orbit of period q or two periodic orbits of period q. Also the renormalization behaviour of fθ with rational rotation number ρθ = p q is studied. MIRAMARE – TRIEST