We consider Thurston maps: branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are sphere trees of bisets. To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer—while centralizers of homeomorphisms are always finitely generated.

中文翻译：

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分支覆盖物 II/V 的算法方面：球体平分和 Thurston 等价的可判定性

我们考虑 Thurston 映射：球体的分支自覆盖与最终周期性临界点，并证明它们之间的 Thurston 等价问题（映射及其临界轨道的连续变形）是可判定的。更准确地说，我们考虑映射类组的作用，通过前组合和后组合，在分支覆盖上，并将它们代数编码为映射类双集。我们展示了保留多曲线的映射的映射类双集如何分解为更小的复杂性的映射类双集，称为小映射类双集。我们根据映射类双集中的共轭和中心化问题来描述球体的分支自覆盖之间的 Thurston 等价决策问题。我们对映射类双集的分解结果将这些决策问题减少到小的映射类双集；它们对应于有理映射、同胚和被环面内同态双重覆盖的映射，它们的共轭和中心化问题分别在复分析、群论和线性代数方面是可解的。分枝覆盖物本身也被编码成双组，具有基本组的动作。我们描述了由拓扑球体之间的分支覆盖产生的那些二分集，并将这种对应关系扩展到具有多曲线的球体之间的映射，其代数对应物是二分集的球树。为了说明 Thurston 映射和同胚之间的区别，