Algorithmic aspects of branched coverings IV/V. Expanding maps

Bartholdi, Laurent; Dudko, Dzmitry

doi:10.1090/tran/7199

Algorithmic aspects of branched coverings IV/V. Expanding maps
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by Laurent Bartholdi and Dzmitry Dudko PDF

Trans. Amer. Math. Soc. 370 (2018), 7679-7714

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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