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- In his Master's thesis, he proved a conjectured of Fatou from 1920 by showing that a rational function of degree has at most nonrepelling periodic cycles.
- He proved that the boundary of the Mandelbrot set has Hausdorff dimension two, confirming a conjecture stated by Mandelbrot and Milnor.
More recent results of Shishikura include
- (in joint work with Kisaka) the existence of a transcendental entire function with a doubly connected wandering domain, answering a question of Baker from 1985;
- (in joint work with Inou) 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.
One of the main tools pioneered by Shishikura and used throughout his work is that of quasiconformal surgery.
- 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.
- P. Fatou, Sur les équations fonctionelles, Bull. Soc. Math. Fr., 1920
- M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 1–29.
- 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 (preprint)
- 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/
- Faculty home page at Kyōto University