# Who proved that the Mandelbrot set's Julia sets are locally connected?

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.)

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.

• So are you saying that the local connectedness of each Julia set with parameter belonging to M implies the MLC conjecture? Aug 20 '19 at 6:01
• @AlgebraicsAnonymous No, I am not saying this, because this is wrong. Aug 20 '19 at 12:09
• @AlexandreEremenko Well, then how is your answer related to the original question? Aug 30 '19 at 7:59
• @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. Dec 29 '19 at 3:44
• arxiv.org/abs/1709.09869