Mitsuhiro Shishikura

Mitsuhiro Shishikura (宍倉 光広, Shishikura Mitsuhiro; born November 27, 1960) is a Japanese mathematician working in the field of complex dynamics. He is professor at Kyoto University in Japan.

Mitsuhiro Shishikura

Shishikura became internationally recognized^[1] for two of his earliest contributions, both of which solved long-standing open problems.

• In his Master's thesis, he proved a conjecture of Fatou from 1920^[2] by showing that a rational function of degree ${\displaystyle d\,}$ has at most ${\displaystyle 2d-2\,}$ nonrepelling periodic cycles.^[3]
• He proved^[4] that the boundary of the Mandelbrot set has Hausdorff dimension two, confirming a conjecture stated by Mandelbrot^[5] and Milnor.^[6]

For his results, he was awarded the Salem Prize in 1992, and the Iyanaga Spring Prize of the Mathematical Society of Japan in 1995.

More recent results of Shishikura include

• (in joint work with Kisaka^[7]) the existence of a transcendental entire function with a doubly connected wandering domain, answering a question of Baker from 1985;^[8]
• (in joint work with Inou^[9]) a study of near-parabolic renormalization which is essential in Buff and Chéritat's recent proof of the existence of polynomial Julia sets of positive planar Lebesgue measure.
• (in joint work with Cheraghi) A proof of the local connectivity of the Mandelbrot set at some infinitely satellite renormalizable points.^[10]
• (in joint work with Yang) A proof of the regularity of the boundaries of the high type Siegel disks of quadratic polynomials.^[11]

One of the main tools pioneered by Shishikura and used throughout his work is that of quasiconformal surgery.

His doctoral students include Weixiao Shen.