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), 14371449
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.
 [BCO08]
Alexander
M. Blokh, Clinton
P. Curry, and Lex
G. Oversteegen, Locally connected models for Julia sets, Adv.
Math. 226 (2011), no. 2, 1621–1661. MR 2737795
(2012d:37106), 10.1016/j.aim.2010.08.011
 [BMO07]
Alexander
Blokh, Michał
Misiurewicz, and Lex
Oversteegen, Planar finitely Suslinian
compacta, Proc. Amer. Math. Soc.
135 (2007), no. 11, 3755–3764 (electronic). MR 2336592
(2009b:54036), 10.1090/S0002993907089538
 [BO04]
Alexander
Blokh and Lex
Oversteegen, Backward stability for polynomial maps
with locally connected Julia sets, Trans. Amer.
Math. Soc. 356 (2004), no. 1, 119–133 (electronic). MR 2020026
(2005c:37081), 10.1090/S0002994703034159
 [DH85]
Adrien
Douady and John
Hamal Hubbard, On the dynamics of polynomiallike mappings,
Ann. Sci. École Norm. Sup. (4) 18 (1985),
no. 2, 287–343. MR 816367
(87f:58083)
 [FS67]
R.
W. FitzGerald and P.
M. Swingle, Core decomposition of continua, Fund. Math.
61 (1967), 33–50. MR 0224063
(36 #7110)
 [GM93]
Lisa
R. Goldberg and John
Milnor, Fixed points of polynomial maps. II. Fixed point
portraits, Ann. Sci. École Norm. Sup. (4) 26
(1993), no. 1, 51–98. MR 1209913
(95d:58107)
 [Kiw04]
Jan
Kiwi, ℝeal laminations and the topological dynamics of
complex polynomials, Adv. Math. 184 (2004),
no. 2, 207–267. MR 2054016
(2005b:37094), 10.1016/S00018708(03)001440
 [KS06]
Oleg
Kozlovski and Sebastian
van Strien, Local connectivity and quasiconformal rigidity of
nonrenormalizable polynomials, Proc. Lond. Math. Soc. (3)
99 (2009), no. 2, 275–296. MR 2533666
(2011a:37096), 10.1112/plms/pdn055
 [Kur66]
K.
Kuratowski, Topology. Vol. I, New edition, revised and
augmented. Translated from the French by J. Jaworowski, Academic Press, New
YorkLondon; Państwowe Wydawnictwo Naukowe, Warsaw, 1966. MR 0217751
(36 #840)
 [LP96]
G.
Levin and F.
Przytycki, External rays to periodic points, Israel J. Math.
94 (1996), 29–57. MR 1394566
(97d:58164), 10.1007/BF02762696
 [Moo62]
R.
L. Moore, Foundations of point set theory, Revised edition.
American Mathematical Society Colloquium Publications, Vol. XIII, American
Mathematical Society, Providence, R.I., 1962. MR 0150722
(27 #709)
 [Mun00]
J. Munkres, Topology, 2nd edition, Prentice Hall, 2000.
 [Nad92]
Sam
B. Nadler Jr., Continuum theory, Monographs and Textbooks in
Pure and Applied Mathematics, vol. 158, Marcel Dekker, Inc., New York,
1992. An introduction. MR 1192552
(93m:54002)
 [QY06]
WeiYuan
Qiu and YongCheng
Yin, Proof of the BrannerHubbard conjecture on Cantor Julia
sets, Sci. China Ser. A 52 (2009), no. 1,
45–65. MR
2471515 (2009j:37074), 10.1007/s1142500801789
 [Sto56]
S.
Stoïlow, Leçons sur les principes topologiques de la
théorie des fonctions analytiques. Deuxième édition,
augmentée de notes sur les fonctions analytiques et leurs surfaces
de Riemann, GauthierVillars, Paris, 1956 (French). MR 0082545
(18,568b)
 [BCO08]
 A. Blokh, C. Curry, L. Oversteegen, Locally connected models for Julia sets, Adv. Math. 226 (2011), 16211661. MR 2737795
 [BMO07]
 Alexander Blokh, Michał Misiurewicz, and Lex Oversteegen, Planar finitely Suslinian compacta, Proc. Amer. Math. Soc. 135 (2007), no. 11, 37553764. MR 2336592 (2009b:54036)
 [BO04]
 Alexander Blokh and Lex Oversteegen, Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc. 356 (2004), no. 1, 119133. MR 2020026 (2005c:37081)
 [DH85]
 A. Douady, J. H. Hubbard, On the dynamics of polynomiallike mappings, Ann. Sci. École Norm. Sup. (4) 18 (1985), 287343. MR 816367 (87f:58083)
 [FS67]
 R. W. FitzGerald and P. M. Swingle, Core decomposition of continua, Fund. Math. 61 (1967), 3350. MR 0224063 (36:7110)
 [GM93]
 L. Goldberg, J. Milnor, Fixed points of polynomial maps. II. Fixed point portraits, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 1, 5198. MR 1209913 (95d:58107)
 [Kiw04]
 Jan Kiwi, Real laminations and the topological dynamics of complex polynomials, Adv. Math. 184 (2004), no. 2, 207267. MR 2054016 (2005b:37094)
 [KS06]
 O. Kozlovski, S. van Strien, Local connectivity and quasiconformal rigidity of nonrenormalizable polynomials, Proc. Lond. Math. Soc. (3) 99 (2009), 275296. MR 2533666 (2011a:37096)
 [Kur66]
 K. Kuratowski, Topology, Volume 1, Academic Press, New York, 1966. MR 0217751 (36:840)
 [LP96]
 G. Levin, F. Przytycki, External rays to periodic points, Israel J. Math. 94 (1996), 2957. MR 1394566 (97d:58164)
 [Moo62]
 R. L. Moore, Foundations of point set theory. Revised edition, AMS Colloquium Publications, 13, Amer. Math. Soc., Providence, RI, 1962. MR 0150722 (27:709)
 [Mun00]
 J. Munkres, Topology, 2nd edition, Prentice Hall, 2000.
 [Nad92]
 Sam B. Nadler, Jr., Continuum theory, Monographs and Textbooks in Pure and Applied Mathematics, vol. 158, Marcel Dekker Inc., New York, 1992. MR 1192552 (93m:54002)
 [QY06]
 W. Qiu, Y. Yin, Proof of the BrannerHubbard conjecture on Cantor Julia sets, Sci. China Ser. A 52 (2009), no. 1, 4565. MR 2471515 (2009j:37074)
 [Sto56]
 S. Stoilow, Leçons sur les principes topologique de la théorie des fonctions analytique, 2nd edition, GauthierVillers, Paris, 1956. MR 0082545 (18:568b)
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Additional Information
Alexander Blokh
Affiliation:
Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 352941170
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 352941170
Email:
overstee@math.uab.edu
DOI:
http://dx.doi.org/10.1090/S000299392012116077
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 DMS0901038
The second author was partially supported by NSF grant DMS0353825.
The third author was partially supported by NSF grant DMS0906316
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.
