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.)
 A: If you mean 


*

*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).

*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).

*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.

A: 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.
