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Conformal Geometry and Dynamics

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Some rational maps whose Julia sets are not locally connected

Author: P. Roesch
Journal: Conform. Geom. Dyn. 10 (2006), 125-135
MSC (2000): Primary 37F50; Secondary 37F10
Published electronically: July 6, 2006
MathSciNet review: 2237276
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Abstract: We describe examples of rational maps which are not topologically conjugate to a polynomial and whose Julia sets are connected but not locally connected.

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  • [AnMa] J. ANDERSON, B. MASKIT, On the local connectivity of limit set of Kleinian groups, Complex Variables Theory Appl. 31 (1996), 177-183. MR 1423249 (98a:30055)
  • [DoHu1] A. DOUADY, J. H. HUBBARD, Etude dynamique des polynômes complexes, Publications mathématiques d'Orsay 1984.
  • [DoHu2] A. DOUADY, J. H. HUBBARD, On the dynamics of polynomial-like mappings, Ann. scient. Éc. Norm. Sup. 18 (1985), 287-343. MR 0816367 (87f:58083)
  • [Do] A. DOUADY, Disques de Siegel et anneaux de Herman, Séminaire Bourbaki, 39 (1986-1987) n$ ^\circ$ 677, 151-172. MR 0936853 (89g:30049)
  • [Gh] E. GHYS, Transformations holomorphes au voisinage d'une courbe de Jordan, C.R.Acad. Sc. Paris, t.298 (1984), 385-388. MR 0748928 (86a:58081)
  • [He] M. HERMAN, Conjugaison quasi-symétrique des difféomorphismes du cercle à des rotations et applications aux disques singuliers de Siegel, I., manuscript, http://www.
  • [Mi1] J. MILNOR, Dynamics in One Complex Variable, Vieweg 1999, 2nd edition 2000. MR 1721240 (2002i:37057)
  • [Mi2] J. MILNOR, Local Connectivity of Julia Sets: Expository Lectures, pp. 67-116 of ``The Mandelbrot set, Theme and Variations'' ed.: Tan Lei, LMS Lecture Note Series 274 , Cambr. U. Press 2000. MR 1765085 (2001b:37073)
  • [Min] Y. MINSKY, On rigidity, limit sets, and end invariants of hyperbolic $ 3$-manifolds, J. Amer. Math. Soc. 7 (1994) no. 3, 539-588. MR 1257060 (94m:57029)
  • [McM1] C. MCMULLEN, Complex Dynamics and Renormalization, Annals of Mathematics Studies, Princeton University Press, Princeton 1994. MR 1312365 (96b:58097)
  • [McM2] C. MCMULLEN, Local connectivity, Kleinian groups, and geodesics on the blowup of the torus, Invent. Math. 146 (2001), 35-91. MR 1859018 (2004e:30068)
  • [Ro] P. ROESCH, On local connectivity for the Julia set of rational maps, Ann. Math. (To appear).
  • [So] D. SORENSEN, Describing quadratic Cremer point polynomials by parabolic perturbations, Ergodic Theory Dynam. Systems 18 (1998), no. 3, 739-758. MR 1631760 (99e:58146)
  • [Su] D. SULLIVAN, Quasiconformal homeomorphisms and dynamics I, solution of the Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401-418. MR 0819553 (87i:58103)
  • [Sh] M. SHISHIKURA, The connectivity of the Julia set of rational maps and Fixed points, Preprint IHES, Bures-sur-Yvette, 1992.
  • [TaYi] L. TAN, Y. YIN, Local connectivity of the Julia set for geometrically finite rational maps, Science in China (Serie A) 39 (1996), 39-47. MR 1397233 (97g:58142)
  • [Wh] G. WHYBURN, Analytic topology, AMS Colloquium Publications 28, 1942. MR 0007095 (4:86b)
  • [Za] S. ZAKERI , In Shahyad, a volume dedicated to Siavash Shahshahani on the occasion of his 60th birthday, 2002.

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Additional Information

P. Roesch
Affiliation: UMR Paul Painleve, University of Lille 1, Cité scientifique - Bâtiment M2, 69655 Villeneuve d’Ascq Cedex, France

Received by editor(s): May 11, 2005
Received by editor(s) in revised form: April 7, 2006
Published electronically: July 6, 2006
Additional Notes: Research partially supported by the Morningside Center of Mathematics in Beijing
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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