Geometry of PCF parameters in spaces of quadratic polynomials

DeMarco, Laura; Mavraki, Niki Myrto

doi:10.2140/ant.2025.19.2163

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Geometry of PCF parameters in spaces of quadratic polynomials

Laura DeMarco and Niki Myrto Mavraki

Vol. 19 (2025), No. 11, 2163–2183

DOI: 10.2140/ant.2025.19.2163

Abstract

We study algebraic relations among postcritically finite (PCF) parameters in the family $f_{c} (z) = z^{2} + c$ . It is known that an algebraic curve in $ℂ^{2}$ contains infinitely many PCF pairs $(c_{1}, c_{2})$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of arbitrary dimension in $ℂ^{n}$ for any $n \geq 2$ . Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in $ℂ^{2}$ and the number of PCF parameters in real algebraic curves in $ℂ$ , depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$ . For $d = 1$ , we prove that there are only finitely many nonspecial lines in $ℂ^{2}$ containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in $ℂ = ℝ^{2}$ containing more than two PCF parameters.

Keywords

Mandelbrot set, postcritically finite maps, quadratic polynomials, special points, unlikely intersections, uniformity, bifurcation measure, equidistribution

Mathematical Subject Classification

Primary: 11G50, 37F46

Milestones

Received: 6 November 2023

Revised: 7 September 2024

Accepted: 18 October 2024

Published: 14 September 2025

Authors

Laura DeMarco
Department of Mathematics Harvard University Cambridge, MA United States

Niki Myrto Mavraki
University of Toronto Toronto, ON Canada

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Vol. 19, No. 11, 2025