**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
*e*^{2πiθ} *z* + *z*^{2}
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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