Galois conjugates of entropies of quadratic maps

The problem of characterizing the algebraic numbers arising from dynamical systems has recently drawn considerable attention. One of the first contexts in which this question makes sense is in the family of real quadratic polynomials; in this case, W. Thurston considered the set of all Galois conjugates of entropies of real quadratic polynomials with finite critical orbit, and found out that this set displays an extremely rich fractal structure. We shall prove that such a set (or rather, its closure) is path-connected and locally connected. Pictures will be provided.