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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Algorithmic aspects of branched coverings IV/V. Expanding maps


Authors: Laurent Bartholdi and Dzmitry Dudko
Journal: Trans. Amer. Math. Soc. 370 (2018), 7679-7714
MSC (2010): Primary 37D20, 37F15, 37F20
DOI: https://doi.org/10.1090/tran/7199
Published electronically: May 30, 2018
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Abstract: Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as Levy-free maps (maps without Levy obstruction) and as maps with contracting biset.

We prove that every Thurston map decomposes along a unique minimal multicurve into homeomorphisms and Levy-free maps, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure.

We apply these results to matings of postcritically finite polynomials, extending a criterion by Mary Rees and Tan Lei: they are expanding if and only if they do not admit a cycle of periodic rays.


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Additional Information

Laurent Bartholdi
Affiliation: École Normale Supérieure, Paris and Mathematisches Institut, Georg-August Universität zu Göttingen
Email: laurent.bartholdi@gmail.com

Dzmitry Dudko
Affiliation: École Normale Supérieure, Paris and Mathematisches Institut, Georg-August Universität zu Göttingen
Email: dzmitry.dudko@gmail.com

DOI: https://doi.org/10.1090/tran/7199
Received by editor(s): October 11, 2016
Received by editor(s) in revised form: January 25, 2017
Published electronically: May 30, 2018
Additional Notes: This work was partially supported by ANR grant ANR-14-ACHN-0018-01 and DFG grant BA4197/6-1
Article copyright: © Copyright 2018 by the authors

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