Mitsuhiro Shishikura

Japanese mathematician (born 1960)

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.

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 $d\,$ has at most $2d-2\,$ 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.

References

^ This recognition is evidenced e.g. by the prizes he received (see below) as well as his invitation as an invited speaker in the Real & Complex Analysis Section of the 1994 International Congress of Mathematicians; see http://www.mathunion.org/o/ICM/Speakers/SortedByCongress.php.
^ Fatou, P. (1920). "Sur les équations fonctionelles" (PDF). Bull. Soc. Math. Fr. 2: 208–314. doi:10.24033/bsmf.1008.
^ M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.
^ Shishikura, Mitsuhiro (1998). "The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets". Annals of Mathematics. Second Series. 147 (2): 225–267. arXiv:math/9201282. doi:10.2307/121009. JSTOR 121009. MR 1626737.
^ B. Mandelbrot, On the dynamics of iterated maps V: Conjecture that the boundary of the M-set has a fractal dimension equal to 2, in: Chaos, Fractals and Dynamics, Eds. Fischer and Smith, Marcel Dekker, 1985, 235-238
^ J. Milnor, Self-similarity and hairiness in the Mandelbrot set, in: Computers in Geometry and Topology, ed. M. C. Tangora, Lect. Notes in Pure and Appl. Math., Marcel Dekker, Vol. 114 (1989), 211-257
^ M. Kisaka and M. Shishikura, On multiply connected wandering domains of entire functions, in: Transcendental dynamics and complex analysis, London Math. Soc. Lecture Note Ser., 348, Cambridge Univ. Press, Cambridge, 2008, 217–250
^ I. N. Baker, Some entire functions with multiply-connected wandering domains, Ergodic Theory Dynam. Systems 5 (1985), 163-169
^ H. Inou and M. Shishikura, The renormalization of parabolic fixed points and their perturbation, preprint, 2008, http://www.math.kyoto-u.ac.jp/~mitsu/pararenorm/
^ Cheraghi, Davoud; Shishikura, Mitsuhiro (2015). "Satellite renormalization of quadratic polynomials". arXiv:1509.07843 [math.DS].
^ Shishikura, Mitsuhiro; Yang, Fei (2016). "The high type quadratic Siegel disks are Jordan domains". arXiv:1608.04106 [math.DS].