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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]].
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 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>
* 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>{{cite journal |arxiv=math/9201282|last1=Shishikura|first1=Mitsuhiro|title=The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets|year=1991|bibcode=1992math......1282S}})</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}}</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>
Dekker, Vol. 114 (1989), 211-257</ref>


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* ''(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 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 [[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 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 journal |arxiv=1509.07843 |last1=Cheraghi |first1=Davoud |last2=Shishikura |first2=Mitsuhiro |title=Satellite renormalization of quadratic polynomials |year=2015 }}</ref>
* ''(in joint work with Cheraghi) A proof of the local connectivity of the [[Mandelbrot set]] at some infinitely satellite renormalizable points.<ref>{{cite arxiv |arxiv=1509.07843 |last1=Cheraghi |first1=Davoud |last2=Shishikura |first2=Mitsuhiro |title=Satellite renormalization of quadratic polynomials |year=2015 }}</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 journal |arxiv=1608.04106 |last1=Shishikura |first1=Mitsuhiro |last2=Yang |first2=Fei |title=The high type quadratic Siegel disks are Jordan domains |year=2016 }}</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 |arxiv=1608.04106 |last1=Shishikura |first1=Mitsuhiro |last2=Yang |first2=Fei |title=The high type quadratic Siegel disks are Jordan domains |year=2016 }}</ref>


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

Revision as of 06:48, 18 April 2024

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.

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

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

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