Algorithmic aspects of branched coverings IV/V. Expanding maps

Abstract

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 and as maps with contracting biset.We prove that every Thurston map decomposes along a unique minimal multicurve into Levy-free and finite-order pieces, and this decomposition is algorithmically computable. Each of these pieces admits a geometric structure.We apply these results to matings of post-critically 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.

On the pullback relation on curves induced by a Thurston map

“…This implies that this example has no Levy cycles. Appealing to [BD1], we conclude there is such an example that is expanding with respect to a complete length metric.…”

Iterated monodromy groups of rational functions and periodic points over finite fields

“…Basic properties of IMGs. Throughout this section, let f : C → C be a PCF rational function of degree d ≥ 2 with post-critical set P f (the same construction works any expanding PCF branched cover f : S 2 → S 2 as in [1], but we will not use the extra generality here). Fix a choice of z 0 ∈ C \ P f .…”

Expansion properties for finite subdivision rules II

“…Suppose f is a Böttcher expanding smooth Thurston map, and M is the complement of neighborhoods of the attractors as in the previous subsection. We recall here some facts from [2,Section 4.2].…”

“…Theorem 3.1. Suppose f is a Thurston map whose orbifold O f has signature (2,2,2,2). Furthermore, supposed f is normalized so that it has a lift to the plane of the form F (x) = Ax + b, where A is a 2 × 2 matrix of integers and b is an integral linear combination of the columns of A.…”