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**An algebraic formulation of Thurston’s characterization of rational functions.**
*(English.
French summary)*
Zbl 1272.37025

Let \(f:S^2\to S^2\) be a branched covering of the two-sphere, and let \(P_f:=\bigcup_{n\geq1}f^n\{f'=0\}\) denote its post-critical set. Assume \(P_f\) is finite, in which case \(f\) is called a Thurston map. Two Thurston maps \(f,g\) are called combinatorially equivalent if they are isotopic via an isotopy \(h_t:S^2\setminus P_f\to S^2\setminus P_g\) that is constant on \(P_f\).

Thurston gave in [A. Douady and J. H. Hubbard, Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)] a characterization of those Thurston maps that are combinatorially equivalent to a rational function, in terms of multicurves. A multicurve for \(f\) is a collection \(\Gamma\) of disjoint, closed, non-trivial closed curves on \(S^2\setminus P_f\), none of which surrounds a single puncture. the multicurve \(\Gamma\) is said to be invariant if every \(f\)-preimage of every curve \(\gamma\in\Gamma\) is either homotopic to a curve in \(\Gamma\) or surrounds at most one puncture. The Thurston linear transformation \(\mathcal L_{f,\Gamma}:\mathbb R^\Gamma\to\mathbb R^\Gamma\) is then defined by \[ \mathcal L_{f,\Gamma}(\gamma)=\sum_{\Gamma\ni\gamma'\simeq\delta\subseteq f^{-1}(\gamma)}\frac1{\deg(f:\delta\to\gamma)}\gamma'. \] Then Thurston’s criterion (in the ‘hyperbolic’ case, which is generic) reads: “\(f\) is equivalent to a rational function if and only if the spectral radius of \(\mathcal L_{f,\Gamma}\) is \(<1\) for every invariant multicurve \(\Gamma\)”.

Let \(G\) denote the pure mapping class group of \((S^2,P_f)\). Again following Douady and Hubbard, there is a finite-index subgroup \(H<G\), consisting of mapping classes that lift via \(f\) to mapping classes in \(G\). Denote by \(\phi_f:H\to G\) the corresponding lifting homomorphism. Then every invariant multicurve \(\Gamma\) gives rise to an abelian subgroup \(\mathbb Z^\Gamma\) of \(G\) consisting of products of Dehn twists about the curves in \(\Gamma\). Note that \(\mathbb Z^\Gamma\) is \(\phi_f\)-invariant, and that the action of \(\phi_f\otimes\mathbb R\) on \(\mathbb Z^\Gamma\otimes\mathbb R\) is naturally conjugate to \(\mathcal L_{f,\Gamma}\).

The author then arrives at the following algebraic characterization of rational maps: “\(f\) is equivalent to a rational function if and only if for every \(\phi_f\)-invariant abelian subgroup \(A\) of \(G\) the spectral radius of the restriction of \(\phi_f\) to \(A\) is \(<1\)”.

He then uses the same grouptheoretico-dynamical techniques to study the dynamics of the “pull-back” function on multicurves. The situation is radically different from that of surface homeomorphisms (when curves, under pullback, tend to become more and more complicated).

He shows that, if \(\phi_f\) is contracting at large scale for the word metric, then the pull-back function on multicurves has a finite global attractor. This is, in particular, the case if \(f\) is a critically finite quadratic polynomial. If \(f\) is merely a rational map (again with ‘hyperbolic orbifold’), he shows that there are finitely many completely invariant multicurves.

He then computes explicitly the attractor for all quadratic polynomials with three finite post-critical points.

Thurston gave in [A. Douady and J. H. Hubbard, Acta Math. 171, No. 2, 263–297 (1993; Zbl 0806.30027)] a characterization of those Thurston maps that are combinatorially equivalent to a rational function, in terms of multicurves. A multicurve for \(f\) is a collection \(\Gamma\) of disjoint, closed, non-trivial closed curves on \(S^2\setminus P_f\), none of which surrounds a single puncture. the multicurve \(\Gamma\) is said to be invariant if every \(f\)-preimage of every curve \(\gamma\in\Gamma\) is either homotopic to a curve in \(\Gamma\) or surrounds at most one puncture. The Thurston linear transformation \(\mathcal L_{f,\Gamma}:\mathbb R^\Gamma\to\mathbb R^\Gamma\) is then defined by \[ \mathcal L_{f,\Gamma}(\gamma)=\sum_{\Gamma\ni\gamma'\simeq\delta\subseteq f^{-1}(\gamma)}\frac1{\deg(f:\delta\to\gamma)}\gamma'. \] Then Thurston’s criterion (in the ‘hyperbolic’ case, which is generic) reads: “\(f\) is equivalent to a rational function if and only if the spectral radius of \(\mathcal L_{f,\Gamma}\) is \(<1\) for every invariant multicurve \(\Gamma\)”.

Let \(G\) denote the pure mapping class group of \((S^2,P_f)\). Again following Douady and Hubbard, there is a finite-index subgroup \(H<G\), consisting of mapping classes that lift via \(f\) to mapping classes in \(G\). Denote by \(\phi_f:H\to G\) the corresponding lifting homomorphism. Then every invariant multicurve \(\Gamma\) gives rise to an abelian subgroup \(\mathbb Z^\Gamma\) of \(G\) consisting of products of Dehn twists about the curves in \(\Gamma\). Note that \(\mathbb Z^\Gamma\) is \(\phi_f\)-invariant, and that the action of \(\phi_f\otimes\mathbb R\) on \(\mathbb Z^\Gamma\otimes\mathbb R\) is naturally conjugate to \(\mathcal L_{f,\Gamma}\).

The author then arrives at the following algebraic characterization of rational maps: “\(f\) is equivalent to a rational function if and only if for every \(\phi_f\)-invariant abelian subgroup \(A\) of \(G\) the spectral radius of the restriction of \(\phi_f\) to \(A\) is \(<1\)”.

He then uses the same grouptheoretico-dynamical techniques to study the dynamics of the “pull-back” function on multicurves. The situation is radically different from that of surface homeomorphisms (when curves, under pullback, tend to become more and more complicated).

He shows that, if \(\phi_f\) is contracting at large scale for the word metric, then the pull-back function on multicurves has a finite global attractor. This is, in particular, the case if \(f\) is a critically finite quadratic polynomial. If \(f\) is merely a rational map (again with ‘hyperbolic orbifold’), he shows that there are finitely many completely invariant multicurves.

He then computes explicitly the attractor for all quadratic polynomials with three finite post-critical points.

Reviewer: Laurent Bartholdi (Göttingen)

### MSC:

37F20 | Combinatorics and topology in relation with holomorphic dynamical systems |

### Keywords:

Thurston map; multicurve; obstruction; rational map; skinning map; mapping class group; virtual endomorphism### Citations:

Zbl 0806.30027
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\textit{K. M. Pilgrim}, Ann. Fac. Sci. Toulouse, Math. (6) 21, No. 5, 1033--1068 (2012; Zbl 1272.37025)

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