September  2017, 37(9): 5049-5063. doi: 10.3934/dcds.2017218

A characterization of Sierpiński carpet rational maps

1. 

Mathematical School of Sichuan University, Chengdu 610065, China

2. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

3. 

Guosen Securities Co., Ltd. Postdoctoral Workstation, Shenzhen 518001, China

* Corresponding author

Received  March 2016 Revised  April 2017 Published  June 2017

In this paper we prove that a postcritically finite rational map with non-empty Fatou set is Thurston equivalent to an expanding Thurston map if and only if its Julia set is homeomorphic to the standard Sierpiński carpet.

Citation: Yan Gao, Jinsong Zeng, Suo Zhao. A characterization of Sierpiński carpet rational maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 5049-5063. doi: 10.3934/dcds.2017218
References:
[1]

M. Bonk and D. Meyer, Expanding Thurston Maps, available from https://sites.google.com/site/dmeyersite/publications. Google Scholar

[2]

B. BonkM. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński Carpet Julia Sets, Adv. Math., 301 (2016), 383-422.  doi: 10.1016/j.aim.2016.06.007.  Google Scholar

[3]

J. W. CannonW. J. FloydR. Kenyon and W.R. Parry, Constructing rational maps from subdivision rules, Conform. Geom Dyn., 7 (2003), 76-102.  doi: 10.1090/S1088-4173-03-00082-1.  Google Scholar

[4]

G. Cui and Y. Gao, Wandering continuum for rational maps, Discrete and Continuous Dynamical System, 36 (2016), 1321-1329.   Google Scholar

[5]

G. CuiW. Peng and L. Tan, On a theorem of Rees-Shishikura, Annales de la Faculté des Sciences de Toulouse, 21 (2012), 981-993.  doi: 10.5802/afst.1359.  Google Scholar

[6]

G. CuiW. Peng and L. Tan, Renormalizations and wandering Jordan curves of rational maps, Communications in Mathematical Physics, 344 (2016), 67-115.  doi: 10.1007/s00220-016-2623-x.  Google Scholar

[7]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297.  doi: 10.1007/BF02392534.  Google Scholar

[8]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes,Ⅰ,Ⅱ, Orsay: Publ. Math. Orsay, 1984.  Google Scholar

[9]

D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 115 (1966), 83-107.  doi: 10.1007/BF02392203.  Google Scholar

[10]

B. Farb and D. Margalit, A Primer on Mapping Class Group, PMS-49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[11]

Y. Gao, P. Haïssinsky, D. Meyer and J. Zeng, Invariant Jordan curves of Sierpiński carpet rational maps, to appear in Ergodic Theory Dynam. Systems, arXiv: 1511.02457. Google Scholar

[12] O. Lehto, Univalent Functions and Teichmüller Spaces, Springer-Verlag, New York, 1987.  doi: 10.1007/978-1-4613-8652-0.  Google Scholar
[13]

Z. Li, Ergodic Theory of Expanding Thurston Maps, Ph. D thesis, University of California Los Angeles, 2015.  Google Scholar

[14]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.  doi: 10.4310/jdg/1214460037.  Google Scholar

[15]

D. Mayer, Expanding Thurston maps as quotients, preprint, arXiv: 0910.2003v1. Google Scholar

[16]

D. Mayer, Invariant Peano curves of expanding Thurston maps, Acta Math., 210 (2013), 95-171.  doi: 10.1007/s11511-013-0091-0.  Google Scholar

[17]

C. McMullen, The classification of conformal dynamical systems, in Current developments in mathematics, 1995(Cambridge, MA), International Press, 1994,323-360.  Google Scholar

[18]

C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I(Mathematical Sciences Research Institute Publications, Springer, New York, New York, 10(1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[19] J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.   Google Scholar
[20]

J. Milnor, Geometry and dynamics of quadratic rational maps, with an appendix by the author and Tan Lei, Experiment. Math., 2 (1993), 37-83.  doi: 10.1080/10586458.1993.10504267.  Google Scholar

[21]

R. L. Moore, Concerning upper semicontinuous collections of compacta, Trans. Amer. Math. Soc., 27 (1925), 416-428.  doi: 10.1090/S0002-9947-1925-1501320-8.  Google Scholar

[22]

Y. XiaoW. Qiu and Y. Yin, On the dynamics of generalized McMullen maps, Ergod. Th. Dyna. Sys., 34 (2014), 2093-2112.  doi: 10.1017/etds.2013.21.  Google Scholar

[23]

G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math., 45 (1958), 320-324.   Google Scholar

[24]

G. T. Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. ⅩⅩⅧ, AMS, Providence, R. I., 1963.  Google Scholar

show all references

References:
[1]

M. Bonk and D. Meyer, Expanding Thurston Maps, available from https://sites.google.com/site/dmeyersite/publications. Google Scholar

[2]

B. BonkM. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński Carpet Julia Sets, Adv. Math., 301 (2016), 383-422.  doi: 10.1016/j.aim.2016.06.007.  Google Scholar

[3]

J. W. CannonW. J. FloydR. Kenyon and W.R. Parry, Constructing rational maps from subdivision rules, Conform. Geom Dyn., 7 (2003), 76-102.  doi: 10.1090/S1088-4173-03-00082-1.  Google Scholar

[4]

G. Cui and Y. Gao, Wandering continuum for rational maps, Discrete and Continuous Dynamical System, 36 (2016), 1321-1329.   Google Scholar

[5]

G. CuiW. Peng and L. Tan, On a theorem of Rees-Shishikura, Annales de la Faculté des Sciences de Toulouse, 21 (2012), 981-993.  doi: 10.5802/afst.1359.  Google Scholar

[6]

G. CuiW. Peng and L. Tan, Renormalizations and wandering Jordan curves of rational maps, Communications in Mathematical Physics, 344 (2016), 67-115.  doi: 10.1007/s00220-016-2623-x.  Google Scholar

[7]

A. Douady and J. H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math., 171 (1993), 263-297.  doi: 10.1007/BF02392534.  Google Scholar

[8]

A. Douady and J. H. Hubbard, Étude Dynamique Des Polynômes Complexes,Ⅰ,Ⅱ, Orsay: Publ. Math. Orsay, 1984.  Google Scholar

[9]

D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 115 (1966), 83-107.  doi: 10.1007/BF02392203.  Google Scholar

[10]

B. Farb and D. Margalit, A Primer on Mapping Class Group, PMS-49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[11]

Y. Gao, P. Haïssinsky, D. Meyer and J. Zeng, Invariant Jordan curves of Sierpiński carpet rational maps, to appear in Ergodic Theory Dynam. Systems, arXiv: 1511.02457. Google Scholar

[12] O. Lehto, Univalent Functions and Teichmüller Spaces, Springer-Verlag, New York, 1987.  doi: 10.1007/978-1-4613-8652-0.  Google Scholar
[13]

Z. Li, Ergodic Theory of Expanding Thurston Maps, Ph. D thesis, University of California Los Angeles, 2015.  Google Scholar

[14]

M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94.  doi: 10.4310/jdg/1214460037.  Google Scholar

[15]

D. Mayer, Expanding Thurston maps as quotients, preprint, arXiv: 0910.2003v1. Google Scholar

[16]

D. Mayer, Invariant Peano curves of expanding Thurston maps, Acta Math., 210 (2013), 95-171.  doi: 10.1007/s11511-013-0091-0.  Google Scholar

[17]

C. McMullen, The classification of conformal dynamical systems, in Current developments in mathematics, 1995(Cambridge, MA), International Press, 1994,323-360.  Google Scholar

[18]

C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli I(Mathematical Sciences Research Institute Publications, Springer, New York, New York, 10(1988), 31-60. doi: 10.1007/978-1-4613-9602-4_3.  Google Scholar

[19] J. Milnor, Dynamics in One Complex Variable, Princeton University Press, 2006.   Google Scholar
[20]

J. Milnor, Geometry and dynamics of quadratic rational maps, with an appendix by the author and Tan Lei, Experiment. Math., 2 (1993), 37-83.  doi: 10.1080/10586458.1993.10504267.  Google Scholar

[21]

R. L. Moore, Concerning upper semicontinuous collections of compacta, Trans. Amer. Math. Soc., 27 (1925), 416-428.  doi: 10.1090/S0002-9947-1925-1501320-8.  Google Scholar

[22]

Y. XiaoW. Qiu and Y. Yin, On the dynamics of generalized McMullen maps, Ergod. Th. Dyna. Sys., 34 (2014), 2093-2112.  doi: 10.1017/etds.2013.21.  Google Scholar

[23]

G. T. Whyburn, Topological characterization of the Sierpiński curve, Fund. Math., 45 (1958), 320-324.   Google Scholar

[24]

G. T. Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. ⅩⅩⅧ, AMS, Providence, R. I., 1963.  Google Scholar

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