Non-degenerate locally connected models for plane continua and Julia sets

Alexander Blokh; Lex Oversteegen; Vladlen Timorin

doi:10.3934/dcds.2017251

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November 2017, 37(11): 5781-5795. doi: 10.3934/dcds.2017251

Non-degenerate locally connected models for plane continua and Julia sets

Alexander Blokh ^1,, , Lex Oversteegen ^1, and Vladlen Timorin ^2,

1.	Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294-1170, USA
2.	Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva St., 119048 Moscow, Russia

* Corresponding author: Alexander Blokh

Received August 2016 Revised June 2017 Published July 2017

Fund Project: The first named author was partially supported by NSF grant DMS-1201450.
The second named author was partially supported by NSF grant DMS-0906316.
The third named author was partially supported by the Russian Academic Excellence Project '5-100'.

Every plane continuum admits a finest locally connected model. The latter is a locally connected continuum onto which the original continuum projects in a monotone fashion. It may so happen that the finest locally connected model is a singleton. For example, this happens if the original continuum is indecomposable. In this paper, we provide sufficient conditions for the existence of a non-degenerate model depending on the existence of subcontinua with certain properties. Applications to complex polynomial dynamics are discussed.

Keywords: Complex dynamics, Julia set, polynomial-like maps, laminations, continuum.

Mathematics Subject Classification: Primary: 37F10, 37F20; Secondary: 37F50, 54C10, 54F15.

Citation: Alexander Blokh, Lex Oversteegen, Vladlen Timorin. Non-degenerate locally connected models for plane continua and Julia sets. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5781-5795. doi: 10.3934/dcds.2017251

References:

[1]	A. Blokh and L. Oversteegen, Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc., 356 (2004), 119-133. doi: 10.1090/S0002-9947-03-03415-9.
[2]	A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Math, 226 (2011), 1621-1661. doi: 10.1016/j.aim.2010.08.011.
[3]	A. Blokh, C. Curry and L. Oversteegen, Finitely Suslinian models for planar compacta with applications to Julia sets, Proc. Amer. Math. Soc., 141 (2013), 1437-1449. doi: 10.1090/S0002-9939-2012-11607-7.
[4]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, Quadratic-like dynamics of cubic polynomials, Communications in Mathematical Physics, 341 (2016), 733-749. doi: 10.1007/s00220-015-2559-6.
[5]	A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
[6]	B. Branner and J. Hubbard, The iteration of cubic polynomials, Part Ⅰ: The global topology of parameter space, Acta Math., 160 (1988), 143-206. doi: 10.1007/BF02392275.
[7]	H. Cremer, Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163. doi: 10.1007/BF01451586.
[8]	A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup.(4), 18 (1985), 287-343. doi: 10.24033/asens.1491.
[9]	J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0.
[10]	K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608.
[11]	J. Milnor, Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp.

show all references

References:

[1]	A. Blokh and L. Oversteegen, Backward stability for polynomial maps with locally connected Julia sets, Trans. Amer. Math. Soc., 356 (2004), 119-133. doi: 10.1090/S0002-9947-03-03415-9.
[2]	A. Blokh, C. Curry and L. Oversteegen, Locally connected models for Julia sets, Advances in Math, 226 (2011), 1621-1661. doi: 10.1016/j.aim.2010.08.011.
[3]	A. Blokh, C. Curry and L. Oversteegen, Finitely Suslinian models for planar compacta with applications to Julia sets, Proc. Amer. Math. Soc., 141 (2013), 1437-1449. doi: 10.1090/S0002-9939-2012-11607-7.
[4]	A. Blokh, L. Oversteegen, R. Ptacek and V. Timorin, Quadratic-like dynamics of cubic polynomials, Communications in Mathematical Physics, 341 (2016), 733-749. doi: 10.1007/s00220-015-2559-6.
[5]	A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.
[6]	B. Branner and J. Hubbard, The iteration of cubic polynomials, Part Ⅰ: The global topology of parameter space, Acta Math., 160 (1988), 143-206. doi: 10.1007/BF02392275.
[7]	H. Cremer, Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163. doi: 10.1007/BF01451586.
[8]	A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup.(4), 18 (1985), 287-343. doi: 10.24033/asens.1491.
[9]	J. Kiwi, $\mathbb R$eal laminations and the topological dynamics of complex polynomials, Advances in Math., 184 (2004), 207-267. doi: 10.1016/S0001-8708(03)00144-0.
[10]	K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608.
[11]	J. Milnor, Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp.

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