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Critical points on the boundaries of Siegel disks

Author: James T. Rogers
Journal: Bull. Amer. Math. Soc. 32 (1995), 317-321
MSC: Primary 30D05; Secondary 54F15, 58F23
MathSciNet review: 1316499
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Abstract: Let f be a polynomial map of the Riemann sphere of degree at least two. We prove that if f has a Siegel disk G on which the rotation number satisfies a diophantine condition, then the boundary of G contains a critical point.

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  • [B] P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Amer. Math. Soc. (N.S.) 11 (1984), 85-141. MR 741725 (85h:58001)
  • [D1] A. Douady, Systèmes dynamiques holomorphes, Sém. Bourbaki, exp. 599, Astérisque 105-106 (1983), 39-63. MR 728980 (85h:58090)
  • [D2] -, Disques de Siegel et anneaux de Herman, Sém. Bourbaki, exp. 677, Astérisque No. 152-153 (1987), 4, 151-172 (1988). MR 936853 (89g:30049)
  • [DH] A. Douady and J. H. Hubbard, Étude dynamique des complexes (deuxième partie), Publ. Math. Orsay 4 (1985), 1-154. MR 812271 (87f:58072b)
  • [G] E. Ghys, Transformation holomorphe au voisinage d'une courbe de Jordan, C. R. Acad. Sci. Paris Sér. I Math. 289 (1984), 385-388. MR 748928 (86a:58081)
  • [H1] M. R. Herman, Are there critical points on the boundary of singular domains?, Comm. Math. Phys. 99 (1985), 593-612. MR 796014 (86j:58067)
  • [H2] -, Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of $ {\text{C}^{n}}$ over a fixed point, Proc. Eighth Internat. Congr. Math. Phys., World Sci. Publ., Singapore, 1987, pp. 138-184.
  • [L] M. Lyubich, The dynamics of rational transforms: The topological picture, Russian Math. Surveys 41:4 (1986), 43-117. MR 863874 (88g:58094)
  • [M] J. Milnor, Dynamics in one complex variable: Introductory lectures, Preprint #1990/5, Inst. Math. Sci., SUNY-Stony Brook. MR 1721240 (2002i:37057)
  • [R1] J. T. Rogers, Jr., Is the boundary of a Siegel disk a Jordan curve?, Bull. Amer. Math. Soc. (N.S.) 27 (1992), 284-287. MR 1160003 (93g:30009)
  • [R2] -, Singularities in the boundaries of local Siegel disks, Ergodic Theory Dynamical Systems 12 (1992), 803-821. MR 1200345 (93m:58061)
  • [R3] -, Diophantine conditions imply critical points on the boundaries of Siegel disks of polynomials, Comm. Math. Phys. (to appear). MR 1637421 (99g:58107)
  • [Y] J.-C. Yoccoz, Linéarisation des germes de difféomorphismes holomorphes de $ {(C,0)}$, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), 55-58. MR 929279 (89i:58123)

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Keywords: Siegel disk, critical point, Julia set, Fatou set, indecomposable continuum, prime end
Article copyright: © Copyright 1995 American Mathematical Society

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