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Characterizing indecomposable plane continua from their complements


Authors: Clinton P. Curry, John C. Mayer and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 136 (2008), 4045-4055
MSC (2000): Primary 54F15; Secondary 37F20
DOI: https://doi.org/10.1090/S0002-9939-08-09508-7
Published electronically: June 26, 2008
MathSciNet review: 2425746
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Abstract: We show that a plane continuum $ X$ is indecomposable iff $ X$ has a sequence $ (U_n)_{n=1}^\infty$ of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence $ (A_n)_{n=1}^\infty$ of open arcs, with $ A_n \subset U_n$ and $ \overline{A_n}\setminus A_n \subset \partial U_n$, there is a sequence of shadows $ (S_n)_{n=1}^\infty$, where each $ S_n$ is a shadow of $ A_n$, such that $ \lim S_n=X$. Such an open arc divides $ U_n$ into disjoint subdomains $ V_{n,1}$ and $ V_{n,2}$, and a shadow (of $ A_n$) is one of the sets $ \partial V_{n,i}\cap \partial U$.


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Additional Information

Clinton P. Curry
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: clintonc@uab.edu

John C. Mayer
Affiliation: Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294-1170
Email: mayer@math.uab.edu

E. D. Tymchatyn
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0W0
Email: tymchat@math.usask.ca

DOI: https://doi.org/10.1090/S0002-9939-08-09508-7
Keywords: Indecomposable continuum, complementary domain, Julia set, complex dynamics, buried point
Received by editor(s): September 4, 2007
Published electronically: June 26, 2008
Additional Notes: The third author was supported in part by NSERC 0GP005616. We thank the Department of Mathematics and Computer Science at Nipissing University, North Bay, Ontario, for the opportunity to work on this paper in pleasant surroundings at their annual topology workshop.
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society

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