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Invariant graphs of rational maps

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Funds:  Supported by NNSF of China (11271344).
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https://doi.org/10.3969/j.issn.0253-2778.2015.03.001
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  • Author Bio:

    CAI Hongjian, male, born in 1987, PhD candidate. Research field: complex dynamic system.

  • Received Date: 06 December 2014
  • Accepted Date: 10 March 2015
  • Rev Recd Date: 10 March 2015
  • Publish Date: 30 March 2015
    Abstract

  • Abstract

    The problem of invariant graphs for rational maps was studied. It was mainly proved that if f is a post-critically finite rational map, whose post-critical set consists of one superattracting fixed point and two points in the Julia set, and whose Julia set is a Sierpinski curve, then an fk-invariant graph containing the post-critical set of f for each sufficiently large k could be found.

    Abstract

    The problem of invariant graphs for rational maps was studied. It was mainly proved that if f is a post-critically finite rational map, whose post-critical set consists of one superattracting fixed point and two points in the Julia set, and whose Julia set is a Sierpinski curve, then an fk-invariant graph containing the post-critical set of f for each sufficiently large k could be found.
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    • References(13)
  • [1]
    Cannon J W, Floyd W J, Parry W R. Constructing subdivision rules from rational maps[J]. Conform Geom Dyn, 2007, 11: 128-136.
    [2]
    Bonk M, Meyer D. Expanding Thurston maps[DB/OL]. arXiv, 2010: arXiv:1009.3647.
    [3]
    Cui G Z, Peng W J, Tan L. On a theorem of Rees-Shishikura[J]. Annales de la Faculté edes Sciences de Toulouse, 2012, XXI(5): 981-993.
    [4]
    Douady A, Hubbard J H. Etude Dynamique des Polynomes Complexes Ⅰ & Ⅱ[M]. Orsay: Publ Math Orsay, 1984-1985.
    [5]
    Shishikura. On a theorem of M. Rees for matings of polynomials[C]// The Mandelbrot set, Theme and Variations. Cambridge: Cambridge Univ Press, 2000: 289-305.
    [6]
    Moore R L. Concerning upper semicontinuous collections of compacta[J]. Trans Amer Math Soc, 1925, 27: 416-426.
    [7]
    Douady A, Hubbard J H. A proof of Thurstons topological characterization of rational functions[J]. Acta Math, 1993, 171: 263-297.
    [8]
    Kameyama A. On Julia sets of postcritically finite branched coverings. I. Coding of Julia sets[J]. J Math Soc Japan, 2003, 55: 439-454.
    [9]
    Lyubich M Y. The dynamics of rational transforms: The topological picture[J]. Russian Math Surveys, 1986, 41 (4): 43-117.
    [10]
    Milnor J. Dynamics in One Complex Variable[M]. Princeton, NJ: Princeton Univ Press, 2006.
    [11]
    Pilgrim K M. Canonical Thurston obstructions[J]. Adv Math, 2001, 158: 154-168.
    [12]
    Pilgrim K M. Combinations of Complex Dynamical Systems[M]. Berlin/ Heidelberg/ New York: Springer-Verlag, 2003.
    [13]
    Thurston W P. Lecture notes[R]. Duluth, MN: CBMS Conference, University of Minnesota at Duluth, 1983.
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    [1]
    Cannon J W, Floyd W J, Parry W R. Constructing subdivision rules from rational maps[J]. Conform Geom Dyn, 2007, 11: 128-136.
    [2]
    Bonk M, Meyer D. Expanding Thurston maps[DB/OL]. arXiv, 2010: arXiv:1009.3647.
    [3]
    Cui G Z, Peng W J, Tan L. On a theorem of Rees-Shishikura[J]. Annales de la Faculté edes Sciences de Toulouse, 2012, XXI(5): 981-993.
    [4]
    Douady A, Hubbard J H. Etude Dynamique des Polynomes Complexes Ⅰ & Ⅱ[M]. Orsay: Publ Math Orsay, 1984-1985.
    [5]
    Shishikura. On a theorem of M. Rees for matings of polynomials[C]// The Mandelbrot set, Theme and Variations. Cambridge: Cambridge Univ Press, 2000: 289-305.
    [6]
    Moore R L. Concerning upper semicontinuous collections of compacta[J]. Trans Amer Math Soc, 1925, 27: 416-426.
    [7]
    Douady A, Hubbard J H. A proof of Thurstons topological characterization of rational functions[J]. Acta Math, 1993, 171: 263-297.
    [8]
    Kameyama A. On Julia sets of postcritically finite branched coverings. I. Coding of Julia sets[J]. J Math Soc Japan, 2003, 55: 439-454.
    [9]
    Lyubich M Y. The dynamics of rational transforms: The topological picture[J]. Russian Math Surveys, 1986, 41 (4): 43-117.
    [10]
    Milnor J. Dynamics in One Complex Variable[M]. Princeton, NJ: Princeton Univ Press, 2006.
    [11]
    Pilgrim K M. Canonical Thurston obstructions[J]. Adv Math, 2001, 158: 154-168.
    [12]
    Pilgrim K M. Combinations of Complex Dynamical Systems[M]. Berlin/ Heidelberg/ New York: Springer-Verlag, 2003.
    [13]
    Thurston W P. Lecture notes[R]. Duluth, MN: CBMS Conference, University of Minnesota at Duluth, 1983.
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