Difference between limbs and bulbs in Mandelbrot Set

2018-10-23 14:54:18

Taking a look to the picture of the Mandelbrot set, one immediately notice its biggest component which we call the main cardioid. This region is composed by the parameters $c$ for which $p_c$ is hyperbolic when its periodic point is a fixed point (that is whose period is $n=1$). It turns out that these parameters are all of the form

$$

c_{\mu}:=\frac {\mu}2\left(1-\frac {\mu}2\right)

$$

when $\mu$ runs in the open unitary disk $\Delta_1$.

For every rational number $\frac pq$ ($p,q$ coprime), there is a circular shaped "bulb" tangent to the main cardioid at the point $c_{\mu}$ for $\mu=e^{2\pi i\frac pq}$, called the $\frac pq$-bulb (clearly $0\le\frac pq\le1$), consisting of all parameters $c$ whose polynomial $p_c$ admits $q$-periodic points.\

The best current estimate known was proved by Yoccoz in the Hubbard paper Local connectivity of Julia sets and bifurcation loci: three theorems of J.C.Yoccoz, which states that the diameters of the $\frac pq$-limbs tends to zero