In this paper we study the well-ordered Cantor sets in hyperbolic sets on the line and the plane. Examples of such sets occur in circle maps and in areapreserving twist maps. We set up a renormalization scheme employing in both cases the first return map. We prove convergence of this scheme. The convergence implies that the asymptotic geometry of such a well-ordered set with irrational rotation number and their nearby well-ordered orbits is determined by the Lyapunov exponent of this set.