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I'd be greatly interested in a reference to the respective article.

Was it Douady? Julia? Hubbard? Fatou?

Bonus question: Who gave the proof that can be found in the Orsay notes?

EDIT: The question was based upon a misconception on the asker's part: There are points in the Mandelbrot set whose Julia sets are NOT locally connected. Yet, all interior points of the Mandelbrot set have a locally connected Julia set. (There are boundary points of the Mandelbrot set whose Julia set is locally connected.)

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2 Answers 2

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Nobody. This is the principal unsolved problem in the area, which is called MLC (That the Mandelbrot set is locally connected). Two Fields medals were awarded for partial progress in this problem.

About Julia sets, some of them are locally connected, others are not. See

The deep significance of the question of the Mandelbrot set's local connectedness?

for more detail.

Remarks. All proofs in the Orsay notes are due to Douady and Hubbard, unless stated otherwise. That Julia sets corresponding to the interior of the Mandelbrot set are locally connected is also not known. It is only known for parts of this interior: for the main hyperbolic component this was proved by Fatou, and for the rest of hyperbolic components by Douady-Hubbard. But the existence of components of the interior, other than hyperbolic components is not known.

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    $\begingroup$ So are you saying that the local connectedness of each Julia set with parameter belonging to M implies the MLC conjecture? $\endgroup$
    – Cloudscape
    Commented Aug 20, 2019 at 6:01
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    $\begingroup$ @AlgebraicsAnonymous No, I am not saying this, because this is wrong. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 20, 2019 at 12:09
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    $\begingroup$ @AlexandreEremenko Well, then how is your answer related to the original question? $\endgroup$
    – Emil Jeřábek
    Commented Aug 30, 2019 at 7:59
  • $\begingroup$ @AlgebraicsAnonymous: there are functions with non locally connected Julia sets and parameter belonging to the Mandelbrot set. However it is not known whether the M set is locally connected. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 29, 2019 at 3:44
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    $\begingroup$ arxiv.org/abs/1709.09869 $\endgroup$
    – Adam
    Commented Sep 13, 2020 at 7:38
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If you mean

  1. are the Julia sets that correspond to parameter values of the Mandelbrot set connected, then I do believe that both Julia and Fatou proved this (in 1918 and 1916 respectively).
  2. are the Julia sets that correspond to parameters values from the interior of the Mandelbrot set locally connected? The definition of local connectivity is post 1918... This is indeed in the Orsay notes, but Douady and Hubbard were quite coy about who wrote what. Most of the results are joint (and some are by the people in the seminar).
  3. are the Julia sets that correspond to parameters values from the boundary of the Mandelbrot set locally connected? This is false, in the sense that some are, and some aren't. A whole slew of people provided useful results here, Siegel, Cramer, Douady-Hubbard, Yoccoz and more. It is a (pun intended) rather complex story.
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  • $\begingroup$ So are you saying that some Julia sets that correspond to the boundary of the Mandelbrot set are not locally connected? (I think there might have been copy&paste issues.) $\endgroup$
    – Cloudscape
    Commented Aug 20, 2019 at 6:00
  • $\begingroup$ By the theorems in the Orsay notes, this would mean that there are parameters in the boundary of M for which there is no indifferent or attracting cycle to which the orbit of zero converges, and neither a repellent cycle in which it ends up after a finite amount of time. $\endgroup$
    – Cloudscape
    Commented Aug 20, 2019 at 6:10
  • $\begingroup$ Follow-up question: Is the set of those boundary parameters whose Julia set is locally connected dense in the boundary? $\endgroup$
    – Cloudscape
    Commented Aug 20, 2019 at 6:27
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    $\begingroup$ @JacquesCarette The first sentences of 2 and 3 are exactly the same. Is this supposed to be this way? $\endgroup$
    – Robert Furber
    Commented Aug 28, 2019 at 21:13
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    $\begingroup$ @AlgebraicsAnonymous As you might expect, the set of parameters where the Julia set is locally connected is dense in the boundary, as is the set of parameters where it is not locally connected. Indeed, every boundary of every hyperbolic component contains points of both types, and it is well-known that little Mandelbrot copies accumulate at every point of the boundary. [This is overkill, as much less is required, but it may give the conceptually clearest picture.] $\endgroup$
    – Lasse Rempe
    Commented Dec 28, 2019 at 0:34

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