Consider an external ray of the Mandelbrot set passing through a point $c_1$ and landing at a point $c_0$ . By a theorem of Carleson, Jones and Yoccoz, the following conditions are equivalent:

(i) : $0$ is not recurrent under $f_{c_0}: z\mapsto z^2 + c_0$ ;

(ii) : The filled Julia set $K_{c_0}$ is a John dendrite.

Condition (i) can be reformulated as a condition (i') involving $K_{c_1}$ , and more specifically the rays in $\C - K_{c_1}$ descending from the critical points of the potential. Condition (i') makes sense for an arbitrary Dirichlet-regular Cantor set. Following an external ray of the Mandelbrot set is a particular case of the Branner Hubbard compression.

So it is reasonnable to think that, if $K$ is an arbitrary Cantor set satisfying (i') - and some regularity conditions -, the Branner-Hubbard compression leads to a John dendrite. We shall present this conjecture.