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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

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

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

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

[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.

[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.

[23]

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

[24]

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

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

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

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

[14]

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

[15]

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

[16]

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

[17]

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

[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.

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

[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.

[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.

[23]

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

[24]

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

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