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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?

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    $\begingroup$ Have you tried sending this question via e-mail to John Hubbard? I suspect the answer is yes, and John would know where a proof is published. $\endgroup$
    – Ryan Budney
    Commented Jan 21, 2023 at 2:45

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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);

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  • $\begingroup$ What does MLC mean in your answer? $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Jan 21, 2023 at 17:52
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    $\begingroup$ Mandelbrot local connectedness, I believe $\endgroup$
    – Anthony Quas
    Commented Jan 21, 2023 at 18:00
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    $\begingroup$ mathoverflow.net/questions/95701/… $\endgroup$
    – Alexandre Eremenko
    Commented Jan 21, 2023 at 23:48
  • $\begingroup$ Indeed, I suppose it is further true that attraction points of hyperbolic components are dense on the boundary. $\endgroup$
    – Geoffrey Irving
    Commented Jan 23, 2023 at 10:30

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