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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
DOI: https://doi.org/10.1090/S0002-9939-2012-11607-7
Published electronically: September 14, 2012
MathSciNet review: 3008890
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Abstract: A compactum $ X\subset \mathbb{C}$ is unshielded if it coincides with the boundary of the unbounded component of $ \mathbb{C}\setminus X$. Call a compactum $ X$ finitely Suslinian if every collection of pairwise disjoint subcontinua of $ X$ whose diameters are bounded away from zero is finite. We show that any unshielded planar compactum $ X$ admits a topologically unique monotone map $ m_X:X \to X_{FS}$ onto a finitely Suslinian quotient such that any monotone map of $ X$ onto a finitely Suslinian quotient factors through $ m_X$. We call the pair $ (X_{FS},m_X)$ (or, more loosely, $ X_{FS}$) the finest finitely Suslinian model of $ X$.

If $ f:\mathbb{C}\to \mathbb{C}$ is a branched covering map and $ X \subset \mathbb{C}$ is a fully invariant compactum, then the appropriate extension $ M_X$ of $ m_X$ monotonically semiconjugates $ f$ to a branched covering map $ g:\mathbb{C}\to \mathbb{C}$ which serves as a model for $ f$. If $ f$ is a polynomial and $ J_f$ is its Julia set, we show that $ m_X$ (or $ M_X$) can be defined on each component $ Z$ of $ J_f$ individually as the finest monotone map of $ Z$ onto a locally connected continuum.


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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: https://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.

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