Multipliers of periodic orbits in spaces of rational maps

Abstract

Given a polynomial or a rational map f we associate to it 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 \rho\not=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function \rho 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.