RENORMALIZATIONS OF CIRCLE HOMEOMORPHISMS WITH A BREAK POINT

Let f_θ(x) = F_0(x)+ θ (mod1); x (in) S^1; θ (in) [0 , 1] be a family of preserving orientationcircle homeomorphisms with a single break point xb, i.e. with a jump in the first derivative F0at the point x = x_b: Suppose that F′_0(x) is absolutely continuous on [xb; xb + 1] and F′′_0(x) (in)L_α([0; 1]) for some α 1: Consider f_θ with rational rotation number ρ_θ = p/q of rank n, i.e.p/q = [k_1; k_2; :::; k_n]: We prove that for sufficiently large n; the renormalizations of f_θ is close tocertain convex linear-fractional functions in C^1+L1.