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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].

External links

Faculty home page at Kyōto University

[1] 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.

[2] Fatou, P. (1920). "Sur les équations fonctionelles" (PDF). Bull. Soc. Math. Fr. 2: 208–314. doi:10.24033/bsmf.1008.

[3] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.

[4] 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.

[5] 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

[6] 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

[7] 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

[8] I. N. Baker, Some entire functions with multiply-connected wandering domains, Ergodic Theory Dynam. Systems 5 (1985), 163-169

[9] 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/

[10] Cheraghi, Davoud; Shishikura, Mitsuhiro (2015). "Satellite renormalization of quadratic polynomials". arXiv:1509.07843 [math.DS].

[11] Shishikura, Mitsuhiro; Yang, Fei (2016). "The high type quadratic Siegel disks are Jordan domains". arXiv:1608.04106 [math.DS].

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@@ Line 1: / Line 1: @@
+{{Short description|Japanese mathematician (born 1960)}}
 [[File:Mitsuhiro Shishikura.jpg|thumb|220px|Mitsuhiro Shishikura]]
-{{nihongo|'''Mitsuhiro Shishikura'''|宍倉 光広|Shishikura Mitsuhiro|born November 27, 1960}} is a [[Japan]]ese [[mathematician]] working in the field of [[complex dynamics]]. He is professor at [[Kyoto University]] in Japan.
+{{nihongo|'''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<ref>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.</ref> for two of his earliest contributions, both of which solved long-standing [[open problems]].
-* In his Master's thesis, he proved a conjectured of [[Pierre Fatou|Fatou]] from 1920<ref>P. Fatou, ''Sur les &eacute;quations fonctionelles'', Bull. Soc. Math. Fr., 1920</ref> by showing that a [[rational function]] of degree <math>d\,</math> has at most <math>2d-2\,</math> nonrepelling [[periodic cycle]]s.<ref>M. Shishikura, ''On the quasiconformal surgery of rational functions,'' Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.</ref>
+* In his Master's thesis, he proved a conjecture of [[Pierre Fatou|Fatou]] from 1920<ref>{{cite journal |first=P. |last=Fatou |title=Sur les équations fonctionelles |url=http://www.numdam.org/article/BSMF_1919__47__161_0.pdf |journal=Bull. Soc. Math. Fr. |year=1920 |volume=2 |pages=208–314 |doi=10.24033/bsmf.1008 |doi-access=free }}</ref> by showing that a [[rational function]] of degree <math>d\,</math> has at most <math>2d-2\,</math> nonrepelling [[periodic cycle]]s.<ref>M. Shishikura, ''On the quasiconformal surgery of rational functions,'' Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.</ref>
+* He proved<ref>{{cite journal
-* He proved<ref>M. Shishikura, ''The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets'', Ann. of Math. (2) 147 (1998), no. 2, 225–267 ([http://arxiv.org/pdf/math/9201282v1.pdf preprint])</ref>  that the boundary of the [[Mandelbrot set]] has [[Hausdorff dimension]] two, confirming a conjecture stated by [[Benoit Mandelbrot|Mandelbrot]]<ref>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</ref> and [[John Milnor|Milnor]].<ref>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
+ | last = Shishikura | first = Mitsuhiro
+ | arxiv = math/9201282
+ | doi = 10.2307/121009
+ | issue = 2
+ | journal = Annals of Mathematics
+ | mr = 1626737
+ | pages = 225–267
+ | series = Second Series
+ | title = The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
+ | volume = 147
+ | year = 1998| jstor = 121009
+ }}</ref>  that the boundary of the [[Mandelbrot set]] has [[Hausdorff dimension]] two, confirming a conjecture stated by [[Benoit Mandelbrot|Mandelbrot]]<ref>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</ref> and [[John Milnor|Milnor]].<ref>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</ref>
@@ Line 11: / Line 24: @@
 More recent results of Shishikura include
 * ''(in joint work with Kisaka<ref>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</ref>)'' the existence of a [[entire function|transcendental entire function]] with a [[doubly connected]] [[wandering domain]], answering a question of Baker from 1985;<ref>I. N. Baker, ''Some entire functions with multiply-connected wandering domains'', Ergodic Theory Dynam. Systems 5 (1985), 163-169</ref>
-* ''(in joint work with Inou<ref>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/</ref>)'' a study of ''near-parabolic renormalization'' which is essential in Buff and [[Arnaud Chéritat|Chéritat]]'s recent proof of the existence of polynomial [[Julia set]]s of positive planar [[Lebesgue measure]].
+* ''(in joint work with Inou<ref>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/</ref>)'' a study of ''near-parabolic renormalization'' which is essential in [[Xavier Buff| Buff]] and [[Arnaud Chéritat|Chéritat]]'s recent proof of the existence of polynomial [[Julia set]]s 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.<ref>{{cite arXiv |eprint=1509.07843 |last1=Cheraghi |first1=Davoud |last2=Shishikura |first2=Mitsuhiro |title=Satellite renormalization of quadratic polynomials |year=2015 |class=math.DS }}</ref>
+* ''(in joint work with Yang)'' A proof of the regularity of the boundaries of the high type [[Siegel disc|Siegel disks]] of quadratic polynomials.<ref>{{cite arXiv |eprint=1608.04106 |last1=Shishikura |first1=Mitsuhiro |last2=Yang |first2=Fei |title=The high type quadratic Siegel disks are Jordan domains |year=2016 |class=math.DS }}</ref>
 One of the main tools pioneered by Shishikura and used throughout his work is that of [[quasiconformal mapping|quasiconformal]] surgery.
+His doctoral students include [[Weixiao Shen]].
 ==References==
@@ Line 20: / Line 37: @@
 ==External links==
 *[http://www.math.kyoto-u.ac.jp/~mitsu/ Faculty home page] at Kyōto University
+{{authority control}}
 {{DEFAULTSORT:Shishikura, Mitsuhiro}}
 [[Category:1960 births]]
 [[Category:Living people]]
-[[Category:Tokyo Institute of Technology faculty]]
+[[Category:Academic staff of Tokyo Institute of Technology]]
-[[Category:University of Tokyo faculty]]
+[[Category:Academic staff of the University of Tokyo]]
-[[Category:Kyoto University faculty]]
+[[Category:Academic staff of Kyoto University]]
 [[Category:Kyoto University alumni]]
 [[Category:20th-century Japanese mathematicians]]

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