No elliptic limits for quadratic maps

We establish bounds for the multipliers of those periodic orbits of R-mu(z) = z(z + mu)/(1 + <(mu)over bar>z), which have a Poincare rotation number p ig. The bounds are given in terms of pig and the (logarithmic) hororadius o f mu to e(2 pi ip/q). The principal tool is a new construction denoted a 's tar' of an immediate attracting basin. The bounds are used to prove propert ies of the space of Mobius conjugacy classes of quadratic rational maps. Th ese properties are related to the mating and non-mating conjecture for quad ratic polynomials [Ta]. Moreover they are also reminiscent of Chuckrows the orem on the non-existence of elliptic limits of loxodromic elements in quas iconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomor phic maps.