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

Thurston, Dylan

Ann. Math. (2) 192, No. 1, 1-46 (2020).

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)].

Reviewer: Bruno Zimmermann (Trieste)

MSC:

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

Keywords:

complex dynamics; rational maps; elastic graphs; quasiconformal surgery; extremal length; Thurston obstruction

Citations:

Zbl 0806.30027; Zbl 1360.37119

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