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Non-persistently recurrent points, qc-surgery and instability of rational maps with totally disconnected Julia sets


Author: Peter M. Makienko
Journal: Conform. Geom. Dyn. 10 (2006), 197-202
MSC (2000): Primary 37F45; Secondary 37F30
DOI: https://doi.org/10.1090/S1088-4173-06-00142-1
Published electronically: September 6, 2006
MathSciNet review: 2261048
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a rational map with a totally disconnected Julia set $ J(R)$. If the postcritical set on $ J(R)$ contains a non-persistently recurrent (or conical) point, then we show that the map $ R$ cannot be a structurally stable map.


References [Enhancements On Off] (What's this?)

  • [BH] B. Branner and J. Hubbard, The iteration of cubic polynomials. II. Patterns and Parapatterns, Acta Math. 169 (1992, no. 3-4), 229-325. MR 1194004 (94d:30044)
  • [DMNU] M. Denker, R.D. Mauldin, Z Nitecki and M. Urbanski, Conformal measures for rational functions revised, Fund. Math. 157 (1998), 161-173. MR 1636885 (99j:58122)
  • [LM] M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geometry 47 (1997), 17-94. MR 1601430 (98k:58191)
  • [MSS] R. Mané, P. Sad and D. Sullivan, On the dynamic of rational maps, Ann. Sci. Ec. Norm. Sup. 16 (1983), 193-217. MR 0732343 (85j:58089)
  • [MM] C. McMullen, Hausdorff dimension and conformal dynamic, II: Geometrically finite rational maps, Comm. Math. Helv. 75 (2000), 535-593. MR 1789177 (2001m:37089)
  • [MS] C. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics, III: The Teichmuller space of a rational map, Adv. Math. 135 (1998), 351-395. MR 1620850 (99e:58145)
  • [M] J. Milnor, Dynamics in one complex variable. Introductory lectures, Friedr. Vieweg and Sohn, Brunschweig, 1999. MR 1721240 (2002i:37057)
  • [P] F. Przytycki, Conical limit set and Poincaré exponent for iterations of rational functions, Trans. Amer. Math. Soc. 351 (1999), 2081-2099. MR 1615954 (99h:58110)
  • [Sh] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. Ecole Norm. Sup. (4) 20 (1987, no. 1), 1-29. MR 0892140 (88i:58099)
  • [S] D. Sullivan, Quasiconformal homeomorphisms and dynamics I, II, III, Ann. of Math. 2 (1985), 401-418; Acta Math. 155 (1985), 243-260. MR 0806415 (87i:58104)

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Additional Information

Peter M. Makienko
Affiliation: Instituto de Matematicas, Av. Universidad S/N., Col. Lomas de Chamilpa, C.P. 62210, Cuernavaca, Morelos, Mexico

DOI: https://doi.org/10.1090/S1088-4173-06-00142-1
Received by editor(s): June 13, 2005
Published electronically: September 6, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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