# Critical points of the multiplier map for the quadratic family

## TL;DR

This paper develops a numerical method to find critical points of the multiplier function for quadratic polynomials and applies it to compute these points for periods up to 10.

## Contribution

It introduces a new numerical algorithm for locating critical points of the multiplier map in quadratic families, extending computations up to period 10.

## Key findings

- Successfully computed critical points for periods up to 10
- Provided insights into the structure of the multiplier map
- Enhanced understanding of bifurcation phenomena in quadratic dynamics

## Abstract

The multiplier $\lambda_n$ of a periodic orbit of period $n$ can be viewed as a (multiple-valued) algebraic function on the space of all complex quadratic polynomials $p_c(z)=z^2+c$. We provide a numerical algorithm for computing critical points of this function (i.e., points where the derivative of the multiplier with respect to the complex parameter $c$ vanishes). We use this algorithm to compute critical points of $\lambda_n$ up to period $n=10$.