Multipliers of periodic orbits in spaces of rational maps

Given a polynomial or a rational function f we include it in a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier ρ⁄=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function ρ in the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space.

QUASICONFORMAL DEFORMATIONS AND VOLUME GROWTH

We determine when a metric measure space admits a uniformizing density which has Ahlfors regular volume growth. A characterization of uniformizing conformal densities in terms of doubling and a lifting procedure are the key ingredients in our presentation. We also furnish an application characterizing metric doubling measures.

No elliptic limits for quadratic maps

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.