A positive characterization of rational maps. (English) Zbl 1450.37042

A topological branched self-cover of the sphere \(S^2\) consists of a finite set of points \(P \subset S^2\) and a map \(f: (S^2,P) \to (S^2, P)\) which is an orientation-preserving covering map when restricted to the complement of the preimage \(f^{-1}(P)\) of \(P\), so \(f\) is a branched cover such that \(f(P) \subset P\) and \(P\) contains all critical values of \(f\) (in particular, \(P\) contains the post-critical set of \(f\), i.e., the orbit of the critical points under iteration of \(f\)). A source of examples are the post-critically finite (i.e., with finite post-critical set) rational maps \(f(z) = P(z)/Q(z)\) of the Riemann sphere \(\widehat{\mathbb C} = \mathbb C \mathbb P^1\). The basic question considered in the present paper is then the following: when is a topological branched self-cover \(f\) of the sphere equivalent to a post-critically finite rational map?
An answer was given by William Thurston in 1982, obtaining a negative characterization: there is a certain combinatorial object (an annular obstruction) that exists exactly when \(f\) is not equivalent to a rational map (see [A. Douady and J. H. Hubbard, Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)]).
In the present paper, a complementary positive criterion is given: a combinatorial object exists exactly when \(f\) is equivalent to a rational map. 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. As the author notes, compared with the previous result mentioned above, the main result of the present paper makes it easier to prove that a map is rational, by just exhibiting an elastic graph spine \(G\) and a suitable map in the homotopy class of a certain virtual endomorphism \(\phi_G^n\) between such graph spines. In the 12-page introduction, the author gives a careful exposition of the concepts and methods of the present paper which completes a program laid out in a research report by the author [Res. Math. Sci. 3, Paper No. 15, 49 p. (2016; Zbl 1360.37119)].


37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F31 Quasiconformal methods in holomorphic dynamics; quasiconformal dynamics
37E25 Dynamical systems involving maps of trees and graphs
57M10 Covering spaces and low-dimensional topology
57R65 Surgery and handlebodies
Full Text: DOI arXiv


[1] {Fortier Bourque}, Maxime, The holomorphic couch theorem, Invent. Math.. Inventiones Mathematicae, 212, 319-406 (2018) · Zbl 1394.30028
[2] Brezin, Eva; Byrne, Rosemary; Levy, Joshua; Pilgrim, Kevin; Plummer, Kelly, A census of rational maps, Conform. Geom. Dyn.. Conformal Geometry and Dynamics. An Electronic Journal of the Amer. Math. Soc., 4, 35-74 (2000) · Zbl 0989.37038
[3] Buff, Xavier; Cui, Guizhen; Tan, Lei, Teichm\"{u}ller spaces and holomorphic dynamics. Handbook of {T}eichm\"{u}ller Theory. {V}ol. {IV}, IRMA Lect. Math. Theor. Phys., 19, 717-756 (2014) · Zbl 1314.30079
[4] Cannon, J. W.; Floyd, W. J.; Kenyon, R.; Parry, W. R., Constructing rational maps from subdivision rules, Conform. Geom. Dyn.. Conformal Geometry and Dynamics. An Electronic Journal of the Amer. Math. Soc., 7, 76-102 (2003) · Zbl 1077.37037
[5] Carrasco Piaggio, Matias, Conformal dimension and canonical splittings of hyperbolic groups, Geom. Funct. Anal.. Geometric and Functional Analysis, 24, 922-945 (2014) · Zbl 1300.30100
[6] Cui, Guizhen; Jiang, Yunping; Sullivan, Dennis, On geometrically finite branched coverings. {I}. {L}ocally combinatorial attracting. Complex Dynamics and Related Topics: Lectures from the {M}orningside {C}enter of {M}athematics, New Stud. Adv. Math., 5, 1-14 (2003) · Zbl 1201.37068
[7] Cui, Guizhen; Jiang, Yunping; Sullivan, Dennis, On geometrically finite branched coverings. {II}. {R}ealization of rational maps. Complex Dynamics and Related Topics: Lectures from the {M}orningside {C}enter of {M}athematics, New Stud. Adv. Math., 5, 15-29 (2003) · Zbl 1201.37067
[8] Cui, Guizhen; Peng, Wenjuan; Tan, Lei, Renormalizations and wandering {J}ordan curves of rational maps, Comm. Math. Phys.. Communications in Mathematical Physics, 344, 67-115 (2016) · Zbl 1362.37095
[9] Cui, Guizhen; Tan, Lei, A characterization of hyperbolic rational maps, Invent. Math.. Inventiones Mathematicae, 183, 451-516 (2011) · Zbl 1230.37052
[10] Cui, Guizhen; Tan, Lei, Hyperbolic-parabolic deformations of rational maps, Sci. China Math.. Science China. Mathematics, 61, 2157-2220 (2018) · Zbl 1404.37049
[11] Douady, Adrien; Hubbard, John Hamal, On the dynamics of polynomial-like mappings, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 18, 287-343 (1985) · Zbl 0587.30028
[12] Douady, Adrien; Hubbard, John Hamal, A proof of {T}hurston’s topological characterization of rational functions, Acta Math.. Acta Mathematica, 171, 263-297 (1993) · Zbl 0806.30027
[13] Epstein, D. B. A., Curves on {\(2\)}-manifolds and isotopies, Acta Math.. Acta Mathematica, 115, 83-107 (1966) · Zbl 0136.44605
[14] Ha{\"{\i}}ssinsky, Peter; Pilgrim, Kevin M., Thurston obstructions and {A}hlfors regular conformal dimension, J. Math. Pures Appl. (9). Journal de Math\'{e}matiques Pures et Appliqu\'{e}es. Neuvi\`eme S\'{e}rie, 90, 229-241 (2008) · Zbl 1213.30048
[15] Hubbard, John; Schleicher, Dierk; Shishikura, Mitsuhiro, Exponential {T}hurston maps and limits of quadratic differentials, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 22, 77-117 (2009) · Zbl 1206.37026
[16] Ishii, Yutaka; Smillie, John, Homotopy shadowing, Amer. J. Math.. American Journal of Mathematics, 132, 987-1029 (2010) · Zbl 1207.37013
[17] Kahn, Jeremy, A priori bounds for some infinitely renormalizable quadratics. {I}. {B}ounded primitive combinatorics (2006)
[18] Kahn, Jeremy; Pilgrim, K. M.; Thurston, D. P., Conformal surface embeddings and extremal length (2015)
[19] Katsura, Takeshi, A class of {\(C^\ast \)}-algebras generalizing both graph algebras and homeomorphism {\(C^\ast \)}-algebras. {I}. {F}undamental results, Trans. Amer. Math. Soc.. Transactions of the Amer. Math. Soc., 356, 4287-4322 (2004) · Zbl 1049.46039
[20] Nekrashevych, Volodymyr, Self-Similar Groups, Math. Surveys Monogr., 117, xii+231 pp. (2005) · Zbl 1087.20032
[21] Nekrashevych, Volodymyr, Combinatorial models of expanding dynamical systems, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 34, 938-985 (2014) · Zbl 1350.37034
[22] Nekrashevych, Volodymyr, Mating, paper folding, and an endomorphism of {\( \Bbb{PC}^2\)}, Conform. Geom. Dyn.. Conformal Geometry and Dynamics. An Electronic Journal of the Amer. Math. Soc., 20, 303-358 (2016) · Zbl 1408.37082
[23] Pilgrim, K.; Thurston, D. P., Graph energies and {A}hlfors-regular conformal dimension
[24] Putman, A., Answer to {W}ho proved that two homotopic embeddings of one surface in another are isotopic? (2016)
[25] Sullivan, Dennis, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. Riemann Surfaces and Related Topics: {P}roceedings of the 1978 {S}tony {B}rook {C}onference, Ann. of Math. Stud., 97, 465-496 (1981)
[26] Thurston, Dylan P., From rubber bands to rational maps: a research report, Res. Math. Sci.. Research in the Mathematical Sciences, 3, 15-49 (2016) · Zbl 1360.37119
[27] Thurston, Dylan P., Elastic graphs, Forum Math. Sigma. Forum of Mathematics. Sigma, 7, 24-84 (2019) · Zbl 1501.37041
[28] Wang, Xiaoguang, A decomposition theorem for {H}erman maps, Adv. Math.. Advances in Mathematics, 267, 307-359 (2014) · Zbl 1418.37078
[29] Zhang, Gaofei; Jiang, Yunping, Combinatorial characterization of sub-hyperbolic rational maps, Adv. Math.. Advances in Mathematics, 221, 1990-2018 (2009) · Zbl 1190.37051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.