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arXiv 2012.002432020-12math.DS

Levy and Thurston obstructions of finite subdivision rules

Insung Park

备注：: 46 pages, 15 figures; v3: many revisions, in particular introducing train-track structures on the non-expanding spines; v4: Section 9.4 is added

英文摘要：

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. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan 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.

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