Algorithmic aspects of branched coverings
Algorithmic aspects of branched coverings
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DOI：
10.5802/afst.1566
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
L. Bartholdi;Dzmitry Dudko
中科院分区：
文献类型：
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作者：
L. Bartholdi;Dzmitry Dudko
这是关于Thurston地图算法研究的一系列文章的公告和长摘要。我们用群理论对象（称为双集）来描述球体的分支覆盖，并发展了双集分解的理论。我们将任意Thurston映射的正则“Levy”分解引入到同胚、度量展开映射和环面自同态双重覆盖的映射中。同胚分解为有限阶和伪anosov映射，展开映射分解为有理映射。作为结果，我们证明了当两个Thurston映射等价时，它是可决定的。我们还证明了上述分解在理论上和实践中都是可计算的。
This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of decompositions of bisets. We introduce a canonical "Levy" decomposition of an arbitrary Thurston map into homeomorphisms, metrically-expanding maps and maps doubly covered by torus endomorphisms. The homeomorphisms decompose themselves into finite-order and pseudo-Anosov maps, and the expanding maps decompose themselves into rational maps. As an outcome, we prove that it is decidable when two Thurston maps are equivalent. We also show that the decompositions above are computable, both in theory and in practice.