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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.   Google Scholar

[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.  Google Scholar

[7]

H. Cremer, Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163.  doi: 10.1007/BF01451586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608.  Google Scholar

[11]

J. Milnor, Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Blokh and L. Oversteegen, Monotone images of Cremer Julia sets, Houston Journal of Mathematics, 36 (2010), 469-476.   Google Scholar

[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.  Google Scholar

[7]

H. Cremer, Zum Zentrumproblem, Math. Ann., 98 (1928), 151-163.  doi: 10.1007/BF01451586.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

K. Kuratowski, Topology Ⅱ, Academic Press, 1968, New York and London, ⅶ-608.  Google Scholar

[11]

J. Milnor, Dynamics in one Complex Variable, Princeton University Press, Princeton, 2006, ⅷ+304pp.  Google Scholar

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