Finitely Suslinian models for planar compacta with applications to Julia sets

Authors:
Alexander Blokh, Clinton Curry and Lex Oversteegen

Journal:
Proc. Amer. Math. Soc. **141** (2013), 1437-1449

MSC (2010):
Primary 54F15; Secondary 37B45, 37F10, 37F20

Published electronically:
September 14, 2012

MathSciNet review:
3008890

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Abstract: A compactum is **unshielded** if it coincides with the boundary of the unbounded component of . Call a compactum **finitely Suslinian** if every collection of pairwise disjoint subcontinua of whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum admits a topologically unique monotone map onto a finitely Suslinian quotient such that any monotone map of onto a finitely Suslinian quotient factors through . We call the pair (or, more loosely, ) the **finest finitely Suslinian model of **.

If is a branched covering map and is a fully invariant compactum, then the appropriate extension of monotonically semiconjugates to a branched covering map which serves as a model for . If is a polynomial and is its Julia set, we show that (or ) can be defined on each component of individually as the finest monotone map of onto a locally connected continuum.

**[BCO08]**Alexander M. Blokh, Clinton P. Curry, and Lex G. Oversteegen,*Locally connected models for Julia sets*, Adv. Math.**226**(2011), no. 2, 1621–1661. MR**2737795**, 10.1016/j.aim.2010.08.011**[BMO07]**Alexander Blokh, Michał Misiurewicz, and Lex Oversteegen,*Planar finitely Suslinian compacta*, Proc. Amer. Math. Soc.**135**(2007), no. 11, 3755–3764 (electronic). MR**2336592**, 10.1090/S0002-9939-07-08953-8**[BO04]**Alexander Blokh and Lex Oversteegen,*Backward stability for polynomial maps with locally connected Julia sets*, Trans. Amer. Math. Soc.**356**(2004), no. 1, 119–133 (electronic). MR**2020026**, 10.1090/S0002-9947-03-03415-9**[DH85]**Adrien Douady and John Hamal Hubbard,*On the dynamics of polynomial-like mappings*, Ann. Sci. École Norm. Sup. (4)**18**(1985), no. 2, 287–343. MR**816367****[FS67]**R. W. FitzGerald and P. M. Swingle,*Core decomposition of continua*, Fund. Math.**61**(1967), 33–50. MR**0224063****[GM93]**Lisa R. Goldberg and John Milnor,*Fixed points of polynomial maps. II. Fixed point portraits*, Ann. Sci. École Norm. Sup. (4)**26**(1993), no. 1, 51–98. MR**1209913****[Kiw04]**Jan Kiwi,*ℝeal laminations and the topological dynamics of complex polynomials*, Adv. Math.**184**(2004), no. 2, 207–267. MR**2054016**, 10.1016/S0001-8708(03)00144-0**[KS06]**Oleg Kozlovski and Sebastian van Strien,*Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials*, Proc. Lond. Math. Soc. (3)**99**(2009), no. 2, 275–296. MR**2533666**, 10.1112/plms/pdn055**[Kur66]**K. Kuratowski,*Topology. Vol. I*, New edition, revised and augmented. Translated from the French by J. Jaworowski, Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR**0217751****[LP96]**G. Levin and F. Przytycki,*External rays to periodic points*, Israel J. Math.**94**(1996), 29–57. MR**1394566**, 10.1007/BF02762696**[Moo62]**R. L. Moore,*Foundations of point set theory*, Revised edition. American Mathematical Society Colloquium Publications, Vol. XIII, American Mathematical Society, Providence, R.I., 1962. MR**0150722****[Mun00]**J. Munkres,*Topology*, 2nd edition, Prentice Hall, 2000.**[Nad92]**Sam B. Nadler Jr.,*Continuum theory*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York, 1992. An introduction. MR**1192552****[QY06]**WeiYuan Qiu and YongCheng Yin,*Proof of the Branner-Hubbard conjecture on Cantor Julia sets*, Sci. China Ser. A**52**(2009), no. 1, 45–65. MR**2471515**, 10.1007/s11425-008-0178-9**[Sto56]**S. Stoïlow,*Leçons sur les principes topologiques de la théorie des fonctions analytiques. Deuxième édition, augmentée de notes sur les fonctions analytiques et leurs surfaces de Riemann*, Gauthier-Villars, Paris, 1956 (French). MR**0082545**

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Additional Information

**Alexander Blokh**

Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Email:
ablokh@math.uab.edu

**Clinton Curry**

Affiliation:
Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794

Email:
clintonc@math.sunysb.edu

**Lex Oversteegen**

Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170

Email:
overstee@math.uab.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2012-11607-7

Keywords:
Continuum,
finitely Suslinian,
locally connected,
monotone map,
Julia set

Received by editor(s):
September 7, 2010

Received by editor(s) in revised form:
August 22, 2011

Published electronically:
September 14, 2012

Additional Notes:
The first author was partially supported by NSF grant DMS-0901038

The second author was partially supported by NSF grant DMS-0353825.

The third author was partially supported by NSF grant DMS-0906316

Communicated by:
Alexander N. Dranishnikov

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.