Present state of the question of Julia sets with positive measure

Until 1990 or so, the dominant conjecture was that the Julia set of a complex rational mapping  f  has measure zero if it is not the whole Riemann sphere. Then attempts were made to construct a counter-example with  f  a polynomial. Recently A. Cheritat has reduced the proof that there is a polynomial of the form eiθ z + z2   with  K(f)  having empty interior but positive measure to renormalization conjectures which seem more accessible. The idea is to obtain θ by alternately approximating rational values by diophantine ones (with bounded type), and diophantine ones by rational ones, so as to lose a small amount of measure for K(f) in the process.

Adrien Douady

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