## 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.

### MSC:

 37F20 Combinatorics and topology in relation with holomorphic dynamical systems

Zbl 0806.30027
Full Text:

### References:

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