No elliptic limits for quadratic maps

Abstract

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic maps.