Let $M$ be the Mandelbrot set.


**Question:** Is the interior of $M$ dense in $M$?


My intuition is that this is true, and moreover that hyperbolic components of the interior are dense in $M$ as well, and moreover that this is known (as it is not very close to the Hyperbolicity Conjecture and thus not too hard). Is that right?



Does the Mandelbrot set have dense interior?
The answer is positive and this is not difficult (a normal families argument). The boundary
of the Mandelbrot set is the set of $J$-instability. Every point $c\_0$ of this set is a limit of $c\_n$ such that $z\mapsto z^2+c\_n$ has a superattracting cycle. So actually hyperbolic components accumulate
to every boundary point of $M$.


(MLC is related to a much harder statement, Fatou conjecture, that interior of $M$ consists of only hyperbolic components).


Refs: M. Lyubich, Some typical properties of the dynamics of rational mappings, English transl.: Russian Math surveys,
38, 1983, or


M. Lyubich, The dynamics of rational transforms: the topological picture,
Russ. Math. Surv. 41, No. 4, 43-117 (1986);


