For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. To show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan [Combining rational maps and controlling obstructions. Ergod. Th. & Dynam. Sys.18(1) (1998), 221–245] to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

中文翻译：

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Levy 和 Thurston 对有限细分规则的阻碍

对于球体的后临界有限分支覆盖（即有限细分规则的细分图），我们定义了非扩展脊柱，它确定非穷举半可判定算法中 Levy 循环的存在。特别是当有限细分规则具有边缘细分的多项式增长时，算法很快终止，Levy环的存在相当于瑟斯顿障碍的存在。为了证明 Levy 和 Thurston 障碍物之间的等价性，我们推广了 Pilgrim 和 Tan 的弧相交障碍物定理 [结合有理图并控制障碍物。埃尔戈德。 Th。 &动态。系统。18(1) (1998), 221–245] 到图相交阻塞定理。作为推论，我们证明对于一对后临界有限多项式，如果至少一个多项式的核心熵为零，那么当且仅当配对存在瑟斯顿障碍时，它们的配对才具有 Levy 循环。