# Equidistribution of critical points of the multipliers in the quadratic family

## TL;DR

This paper proves that critical points of period n multipliers in quadratic polynomials become evenly distributed along the Mandelbrot set boundary as n increases, revealing a deep connection between dynamics and complex geometry.

## Contribution

It establishes the equidistribution of critical points of multipliers for quadratic polynomials, linking dynamical properties to geometric distribution on the Mandelbrot set boundary.

## Key findings

- Critical points of period n multipliers equidistribute on the Mandelbrot boundary as n→∞
- The distribution of these points reflects the complex structure of the quadratic family
- The result connects dynamical stability with geometric boundary behavior

## Abstract

A parameter $c_0\in\mathbb C$ in the family of quadratic polynomials $f_c(z)=z^2+c$ is a critical point of a period $n$ multiplier, if the map $f_{c_0}$ has a periodic orbit of period $n$, whose multiplier, viewed as a locally analytic function of $c$, has a vanishing derivative at $c=c_0$. We prove that all critical points of period $n$ multipliers equidistribute on the boundary of the Mandelbrot set, as $n\to\infty$.