Abstract

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 $e2\pi \; i\theta z\; +\; z2$ with $K(f)$ having empty interior but positive measure to renormalization conjectures which seem more accessible.