Counting hyperbolic components in the main molecule

D’Souza, Schinella; Tiozzo, Giulio

doi:10.2140/involve.2025.18.747

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Counting hyperbolic components in the main molecule

Schinella D’Souza and Giulio Tiozzo

Vol. 18 (2025), No. 5, 747–754

DOI: 10.2140/involve.2025.18.747

Abstract

We count the number of hyperbolic components of period $n$ that lie on the main molecule of the Mandelbrot set. We give a formula for how to compute the number of these hyperbolic components of period $n$ in terms of the divisors of $n$ and, in the prime power case, an explicit formula is derived.

Keywords

Mandelbrot set, hyperbolic components, holomorphic dynamics

Mathematical Subject Classification

Primary: 37F20

Secondary: 37F10

Milestones

Received: 14 July 2022

Revised: 14 July 2024

Accepted: 17 July 2024

Published: 13 November 2025

Communicated by Kenneth S. Berenhaut

Authors

Schinella D’Souza
Department of Mathematics University of Michigan Ann Arbor, MI United States

Giulio Tiozzo
Department of Mathematics University of Toronto Toronto, ON Canada

Vol. 18, No. 5, 2025