TITLE: Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set?
QUESTION [16 upvotes]: Let $\mathcal{B}$ denote the boundary of the Mandelbrot set, and let
$\overline{\mathbb{Q}}$ denote the algebraic closure of the rationals.
Further put $\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{B} \cap 
\overline{\mathbb{Q}}$.
Questions:

Is $\mathcal{B}_{\overline{\mathbb{Q}}}$ dense in $\mathcal{B}$
in the sense that for every $z \in \mathcal{B}$ and every
$\varepsilon > 0$, the $\varepsilon$-neighborhood of $z$ has nontrivial 
intersection with $\mathcal{B}_{\overline{\mathbb{Q}}}$?
Given $z \in \mathbb{C}$, let
$\mathcal{B} + z := \{b + z \ | \ b \in \mathcal{B}\}$
denote the translate of $\mathcal{B}$ by $z$.
Is there a $z \in \mathbb{C}$ such that
$(\mathcal{B} + z) \cap \overline{\mathbb{Q}} = \emptyset$?
What are the answers to (1.) and (2.) if we replace
$\overline{\mathbb{Q}}$ by the Gaussian rationals $\mathbb{Q}[i]$
or by the cyclotomic integers $\mathbb{Z}[e^{\frac{2\pi i}{5}}]$?

REPLY [27 votes]: Post-critically pre-periodic quadratic polynomials, i.e. those for which the orbit of the critical point $0$ is pre-periodic, are well-known to be dense in the boundary of the Mandelbrot set. (This is essentially a normality argument.)
Each of these is determined by an algebraic equation. This answers your first question (the question in the title).
EDIT. As pointed out by Malik, post-critically pre-periodic parameters are also called Misiurewicz points. (Unfortunately, the term Misiurewicz is sometime also used, particularly in real dynamics I think, to refer to a larger class of systems.)