Quadratic polynomials, multipliers and equidistribution

TL;DR

This paper investigates how parameters for quadratic maps with specified cycle multipliers distribute asymptotically, revealing a phase transition at a critical growth rate of the multipliers.

Contribution

It characterizes the asymptotic distribution of parameters with prescribed multipliers, connecting growth rates to geometric loci in the Mandelbrot set.

Findings

Parameters with bounded multipliers equidistribute on the Mandelbrot boundary.

Parameters with rapidly growing multipliers equidistribute on specific equipotentials.

Identifies a phase transition at the critical growth rate L = log 2.

Abstract

Given a sequence of complex numbers {\rho}_n, we study the asymptotic distribution of the sets of parameters c {\epsilon} C such that the quadratic maps z^2 +c has a cycle of period n and multiplier {\rho}_n. Assume 1/n.log|{\rho}_n| tends to L. If L {\leq} log 2, they equidistribute on the boundary of the Mandelbrot set. If L > log 2 they equidistribute on the equipotential of the Mandelbrot set of level 2L - 2 log 2.