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Planar finitely Suslinian compacta
Authors:
Alexander Blokh, Michal\ Misiurewicz and Lex Oversteegen
Journal:
Proc. Amer. Math. Soc. 135 (2007), 3755-3764
MSC (2000):
Primary 54F15, 54D05, 37F10
Posted:
August 15, 2007
MathSciNet review:
2336592
Full-text PDF Free Access
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Additional Information
Abstract: We show that a planar unshielded compact set  is finitely Suslinian if and only if there exists a closed set  and a lamination  of  such that  is homeomorphic to  . If  is a continuum, the analogous statement follows from Carathéodory theory and is widely used in polynomial dynamics.
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Alexander
Blokh, Chris
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pp. 253–270. MR 2090774
(2006d:37076)
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Alexander
Blokh, Chris
Cleveland, and Michał
Misiurewicz, Julia sets of expanding polymodials, Ergodic
Theory Dynam. Systems 25 (2005), no. 6,
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(2006h:37071), http://dx.doi.org/10.1017/S0143385705000210
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- 2.
- A. Blokh, C. Cleveland, and M. Misiurewicz, Julia sets of expanding polymodials, Ergodic Theory and Dynam. Syst. 25 (2005), 1691-1718. MR 2183289 (2006h:37071)
- 3.
- A. Blokh and L. Oversteegen, Backward stability for polynomial maps, Trans. Amer. Math. Soc. 356 (2004), 119-133. MR 2020026 (2005c:37081)
- 4.
- R. D. Bourgin and P. L. Renz, Shortest paths in simply connected regions in
 , Adv. Math. 76 (1989), 260-295. MR 1013673 (90k:52021)
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- M. Brown, Sets of constant distance from a planar set, Michigan Math. J. 19 (1972), 321-323. MR 0315714 (47:4263)
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- A. Douady, Descriptions of compact sets in
 , in: Topological methods in modern mathematics, Publish or Perish, (1993), pp. 429-465. MR 1215973 (94g:58185)
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- P. Fabel, ``Shortest'' arcs in closed planar disks vary continuously with the boundary, Top. Appl. 95 (1999), 75-83. MR 1691933 (2000d:58016)
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- K. Kuratowski, Topology II, Academic Press, New York, 1968.
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- C. T. McMullen, Complex dynamics and renormalization, Annals of Mathematical Studies 135, Princeton University Press, Princeton, NJ (1994). MR 1312365 (96b:58097)
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Additional Information
Alexander Blokh
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email:
ablokh@math.uab.edu
Michal\ Misiurewicz
Affiliation:
Department of Mathematical Sciences, IUPUI, 402 N. Blackford Street, Indianapolis, Indiana 46202-3216
Email:
mmisiure@math.iupui.edu
Lex Oversteegen
Affiliation:
Department of Mathematics, University of Alabama in Birmingham, University Station, Birmingham, Alabama 35294-2060
Email:
overstee@math.uab.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08953-8
PII:
S 0002-9939(07)08953-8
Keywords:
Finitely Suslinian,
unshielded,
locally connected,
lamination
Received by editor(s):
January 4, 2006
Received by editor(s) in revised form:
September 8, 2006
Posted:
August 15, 2007
Additional Notes:
The first author was partially supported by NSF grant DMS 0456748
The second author was partially supported by NSF grant DMS 0456526
The third author was partially supported by by NSF grant DMS 0405774
Communicated by:
Alexander N. Dranishnikov
Article copyright:
© Copyright 2007 American Mathematical Society
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