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On dynamics of vertices of locally connected polynomial Julia sets

Author(s): A. Blokh; G. Levin
Journal: Proc. Amer. Math. Soc. 130 (2002), 3219-3230.
MSC (2000): Primary 37F10; Secondary 37E25
Posted: May 29, 2002
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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:

[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]
B. Branner, J. Hubbard, The iteration of cubic polynomials. Part II: Patterns and parapatterns., Act Math. 169 (1992), 229-325. MR 94d:30044

[CJY]
L. Carleson, P. Jones, J.-C. Yoccoz, Julia and John, Bol. Soc. Brasil. Mat. 25 (1994), 1-30. MR 95d:30040

[CL]
E. F. Collingwood, A. J. Lohwater, The Theory of Cluster Sets, Cambridge University Press, Cambridge, 1966. MR 38:325

[Do]
A. Douady, Descriptions of compact sets in ${\mathbb{C} } $, Topological methods in modern mathematics, Publish or Perish, 1993, pp. 429-465. MR 94g:58185

[DH]
A. Douady and J. H. Hubbard, Étude dynamique des polynômes complexes I, II, Publication Mathematiques d'Orsay, vol. 84-2, 1984; vol. 85-04, 1985. MR 87f:58072a; MR 87f:58072b

[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, Publish or Perish, 1993, pp. 467-511. MR 94c:58172

[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, Fundamenta Mathematicae 158 (1998), 97-107. MR 99j:58171

[McM]
C. T. McMullen, Complex dynamics and renormalization, Annals of Mathematical Studies, vol. 135, Princeton University Press, Princeton, NJ, 1994. MR 96b:58097

[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]
D. Sullivan, Conformal dynamical systems, in: Geometric Dynamics, Lecture Notes in Math., vol. 1007, Springer Verlag, NY, 1983, pp. 725-752. MR 85m:58112

[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
Email: ablokh@math.uab.edu

G. Levin
Affiliation: Institute of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem, Israel
Email: levin@math.huji.ac.il

DOI: 10.1090/S0002-9939-02-06698-4
PII: S 0002-9939(02)06698-4
Keywords: Julia set, vertices, laminations, recurrent critical points
Received by editor(s): December 22, 2000
Posted: May 29, 2002
Additional Notes: The first author was partially supported by NSF grant DMS 9970363.
Communicated by: Michael Handel
Copyright of article: Copyright 2002, American Mathematical Society

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