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On dynamics of vertices of locally connected polynomial Julia sets

Authors: A. Blokh and G. Levin
Journal: Proc. Amer. Math. Soc. 130 (2002), 3219-3230
MSC (2000): Primary 37F10; Secondary 37E25
Published electronically: May 29, 2002
MathSciNet review: 1912999
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Abstract: Let $P$ be a polynomial whose Julia set $J$ is locally connected. Then a non-preperiodic non-precritical vertex of $J$must have the limit set which coincides with the limit set of an appropriately chosen recurrent critical point of $P$. In particular, if all critical points of $P$ are non-recurrent then all vertices of $J$ are preperiodic or precritical.

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

  • [BL1] A. Blokh, G. Levin, An inequality for laminations, Julia sets and ``growing trees'', Erg. Th. and Dyn. Syst. 22 (2002), 63-97.
  • [BL2] -, Growing trees, laminations and the dynamics on the Julia set, IHES Preprint IHES/M/99/77 (1999).
  • [BM] A. Blokh, M. Misiurewicz, Attractors for graph critical rational functions, Trans. Amer. Math. Soc. (to appear).
  • [BO] A. Blokh, l. Oversteegen, Backward stability for polynomial maps with locally connected Julia sets, Preprint, 2000.
  • [BH] Bodil Branner and John H. Hubbard, The iteration of cubic polynomials. II. Patterns and parapatterns, Acta Math. 169 (1992), no. 3-4, 229–325. MR 1194004,
  • [CJY] Lennart Carleson, Peter W. Jones, and Jean-Christophe Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. (N.S.) 25 (1994), no. 1, 1–30. MR 1274760,
  • [CL] E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 0231999
  • [Do] Adrien Douady, Descriptions of compact sets in 𝐶, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 429–465. MR 1215973
  • [DH] A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 (French). MR 762431
  • [F] P. Fatou, Sur les equations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161-271; vol. 48, 1920, pp. 33-94, 208-314.
  • [H] J. H. Hubbard, Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991) Publish or Perish, Houston, TX, 1993, pp. 467–511. MR 1215974
  • [J] G. Julia, Memoire sur l'iteration des fonctions rationelles, J. Math. Pure Appl. 8 (1919), 47-245.
  • [Ki] J. Kiwi, Rational rays and critical portraits of complex polynomials, SUNY at Stony Brook and IMS Preprint #1997/15.
  • [Kur] C. Kuratowski, Topologie, vol. 2, Warszawa-Wroc\law, 1950. MR 12:517a
  • [L] G. Levin, On backward stability of holomorphic dynamical systems, Fund. Math. 158 (1998), no. 2, 97–107. MR 1656942
  • [McM] Curtis T. McMullen, Complex dynamics and renormalization, Annals of Mathematics Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 1312365
  • [Mi] J. Milnor, Dynamics in One Complex Variable: Introductory Lectures, SUNY at Stony Brook and IMS Preprint #1990/5.
  • [Po] A. Poirier, On post critically finite polynomials. Part two: Hubbard trees, SUNY at Stony Brook and IMS Preprint #1993/7.
  • [Sul] Dennis Sullivan, Conformal dynamical systems, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 725–752. MR 730296,
  • [Th] W. Thurston, The combinatorics of iterated rational maps, Preprint, 1985.

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Additional Information

A. Blokh
Affiliation: Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060

G. Levin
Affiliation: Institute of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem, Israel

Keywords: Julia set, vertices, laminations, recurrent critical points
Received by editor(s): December 22, 2000
Published electronically: May 29, 2002
Additional Notes: The first author was partially supported by NSF grant DMS 9970363.
Communicated by: Michael Handel
Article copyright: © Copyright 2002 American Mathematical Society