Finitely Suslinian models for planar compacta with applications to Julia sets
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- by Alexander Blokh, Clinton Curry and Lex Oversteegen PDF
- Proc. Amer. Math. Soc. 141 (2013), 1437-1449 Request permission
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
- MR Author ID: 196866
- 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
- MR Author ID: 134850
- Email: overstee@math.uab.edu
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3008890