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An algebraic formulation of Thurston's combinatorial equivalence

Author: Kevin M. Pilgrim
Journal: Proc. Amer. Math. Soc. 131 (2003), 3527-3534
MSC (2000): Primary 37F20; Secondary 20F28, 20F36, 20E05
Published electronically: May 7, 2003
MathSciNet review: 1991765
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Abstract: Let $f:S^2 \to S^2$ be an orientation-preserving branched covering for which the set $P_f$ of strict forward orbits of critical points is finite and let $G=\pi_1(S^2-f^{-1}P_f)$. To $f$ we associate an injective endomorphism $\varphi_f$ of the free group $G$, well-defined up to postcomposition with inner automorphisms. We show that two such maps $f,g$ are combinatorially equivalent (in the sense introduced by Thurston for the characterization of rational functions as dynamical systems) if and only if $\varphi_f, \varphi_g$ are conjugate by an element of $\operatorname{Out}(G)$ which is induced by an orientation-preserving homeomorphism.

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Additional Information

Kevin M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-7106

Keywords: Postcritically finite, endomorphism of free group
Received by editor(s): June 20, 2002
Published electronically: May 7, 2003
Additional Notes: This research was supported by Indiana University, Bloomington
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society