# Subsets of the Mandelbrot set

In order to define the Mandelbrot set $$\mathbb M$$, one looks at the sequences $$s = f(0),(f\circ f)(0), (f\circ f \circ f)(0),\ldots,$$ where $$f:\mathbb C \to \mathbb C,~z\mapsto z^2+c$$ for different $$c\in\mathbb C$$. Now if $$c$$ is such that the resulting sequence $$s$$ is bounded, then $$c\in\mathbb M$$, otherwise $$c\notin \mathbb M$$. Does anyone know if there are characterizations or (even better) visualizations of the following related sets? I'd be grateful for a reference.

\begin{aligned} \mathbb M_\ell & :=\{c\in \mathbb C, s\text{ is a periodic sequence with period \ell.}\}\\ \mathbb M_p & := \bigcup_{\ell \in \mathbb N}\mathbb M_{\ell}\\ \mathbb M_{np} & := \mathbb M\setminus \mathbb M_p\\ \mathbb M_{\text{con}} & := \{c \in \mathbb C, s\text{ converges.}\} \end{aligned}

• The sets $$M_l$$ are finite (they have $$2^l$$ elements). They are "in the center" of the connected components of the interior of the Mandelbrot set, and are not on its boundary.
• The set $$M_p$$ is therefore countable, consisting of points all in the interior of $$M$$; but its closure contains the boundary of $$M$$.
• $$M_{np}$$ is too big to have a simple description: after all, you only removed countably many points from $$M$$.
• Presumably, $$M_{con}$$ is exactly the interior of the main cardioid (plus the parameter $$c=\frac{1}{4}$$, together with a countable collection of points corresponding to parameters for which the sequence $$s$$ is eventually constant. Those countable parameters are located at certain tips on the boundary of the Mandelbrot set.
• Thanks a lot for your answer. Could you please explain how you find the cardinality of $M_l$ and what the reason for your guess for $M_{con}$ is?
• mrob.com/pub/muency/enumerationoffeatures.html $f_c^p(0) = 0$ is a degree $2^{p-1}$ polynomial in $c$ so it has $2^{p -1}$ roots. Some of these roots may have a lower period too, maybe you want to count them in only one $M_l$ (with $l$ as low as possible). I'm not sure how to take into account potential multiplicity of the roots. Dec 3, 2019 at 11:40