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.

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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.