I found some proofs online but I don't consider them acceptable.
http://www.math.cornell.edu/~hubbard/OrsayEnglish.pdf there is a "proof" (page 22) that doesn't satisfy me. I don't see why the branches of the inverses are Lipschitz.
http://www.maths.usyd.edu.au/u/UG/IM/MATH2916/r/Long_Essay.pdf (page 5) This proof does not satisfy me AT ALL! Because the construction does not remind me of the construction of the Cantor set, since when I construct the Cantor set i know EXACTLY how much I'm removing at each stage.
https://esc.fnwi.uva.nl/thesis/centraal/files/f358061116.pdf (page 22) This proof is not convincing since it doesn't tell what really is "Mod" and it relies on results that are not proved. This whole paper is full of mistakes and imprecisions.
https://zr9558.files.wordpress.com/2013/11/complex-dynamics-carleson.pdf (page 67) The proof relies on a lot of powerful theorems proved in chapter one. I know that they are useful but the definition of Julia set is hard to understand and different from the one I have.
How come that for such a result that is mentioned everywhere as the "Fundamental Dichotomy" is almost impossible to find a decent proof?