Bifurcation measures and quadratic rational maps

TL;DR

This paper investigates bifurcation phenomena in quadratic rational maps, characterizing when certain parameter curves contain infinitely many postcritically finite maps and analyzing bifurcation measures associated with critical points.

Contribution

It proves that the curve Per_1(λ) contains infinitely many postcritically finite maps only when λ=0, confirming a special case of a conjecture, and analyzes bifurcation measures of critical points.

Findings

Per_1(λ) contains infinitely many postcritically finite maps iff λ=0

Critical points define distinct bifurcation measures on Per_1(λ)

Addresses a special case of a conjecture in complex dynamics

Abstract

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(\lambda)$ for $\lambda$ in $\mathbb{C}$, defined by the condition that each $f\in Per_1(\lambda)$ has a fixed point of multiplier $\lambda$. We prove that the curve $Per_1(\lambda)$ contains infinitely many postcritically-finite maps if and only if $\lambda = 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(\lambda)$.