The size of Mandelbrot bulbs

Abstract

We provide an analytic estimate for the size of the bulbs adjoining the main cardioid of the Mandelbrot set. The bulbs are approximate circles, and are associated with the stability regions in the complex parameter μ-space of period-q orbits of the underlying map $z\to z^2-\mu$. For the (p, q) orbit with winding number p/q, the associated stability bulb is an approximate circle with radius $\frac{1}{q^2}\sin \frac{\pi p}{q}$.