Counting Hyperbolic Components in the Main Molecule

TL;DR

This paper provides a formula to count hyperbolic components of period n on the main molecule of the Mandelbrot set, including explicit formulas for prime power cases, advancing understanding of the set's structure.

Contribution

It introduces a new counting formula for hyperbolic components on the main molecule, including explicit solutions for prime power periods.

Findings

Derived a general formula for counting hyperbolic components of period n

Provided explicit formulas for prime power cases

Enhanced understanding of the Mandelbrot set's structure

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.