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.
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 has at most 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
[edit]- ^ 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].
External links
[edit]- Faculty home page at Kyōto University
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