A positive characterization of rational maps

Abstract

When is a topological branched self-cover of the sphere equivalent to a post-critically finite rational map on $\mathbb {C}\mathbb {P}^1$? William Thurston gave one answer in 1982, giving a negative criterion (an obstruction to a map being rational). We give a complementary, positive criterion: the branched self-cover is equivalent to a rational map if and only if there is an elastic graph spine for the complement of the post-critical set that gets “looser” under backwards iteration.

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      pages = {307--359},
      issn = {0001-8708},
      mrclass = {37F30 (37F10 37F20 37F50)},
      mrnumber = {3269181},
      mrreviewer = {Jasmin Raissy},
      doi = {10.1016/j.aim.2014.09.004},
      url = {https://doi.org/10.1016/j.aim.2014.09.004},
      zblnumber = {1418.37078},
      }
  • [JZ09:SubHyp] Go to document G. Zhang and Y. Jiang, "Combinatorial characterization of sub-hyperbolic rational maps," Adv. Math., vol. 221, iss. 6, pp. 1990-2018, 2009.
    @ARTICLE{JZ09:SubHyp,
      author = {Zhang, Gaofei and Jiang, Yunping},
      title = {Combinatorial characterization of sub-hyperbolic rational maps},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {221},
      year = {2009},
      number = {6},
      pages = {1990--2018},
      issn = {0001-8708},
      mrclass = {37F20 (37F10)},
      mrnumber = {2522834},
      mrreviewer = {David A. Brown},
      doi = {10.1016/j.aim.2009.03.009},
      url = {https://doi.org/10.1016/j.aim.2009.03.009},
      zblnumber = {1190.37051},
      }

Authors

Dylan P. Thurston

Indiana University, Bloomington, IN