Multipliers of periodic orbits in spaces of rational maps

TL;DR

This paper explores the relationship between the multipliers of periodic orbits and the local coordinates in the space of rational maps, using quasiconformal deformation theory to establish new connections.

Contribution

It introduces a new framework linking periodic orbit multipliers to local coordinates in rational map spaces, enhancing understanding of their dynamical structure.

Findings

Multipliers of non-repelling periodic orbits serve as local coordinates.

A simple connection between the dynamical plane and the multiplier function is established.

The proof utilizes quasiconformal deformation theory.

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.