Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

Selinger, Nikita; Yampolsky, Michael

doi:10.1007/s40598-015-0024-4

Cited by 9 publications

(17 citation statements)

References 24 publications

(44 reference statements)

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“…We will prove Theorem 4.4 for maps not doubly covered by torus endomorphisms. The remaining case follows from [12,Theorem 4] or from [20]. The hardest implication in the proof is (4)ñ(1), and will occupy most of this section.…”

Section: Expanding Non-torus Mapsmentioning

confidence: 99%

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Algorithmic aspects of branched coverings IV/V. Expanding maps

Bartholdi

Dudko

2018

Trans. Amer. Math. Soc.

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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.

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Section: Expanding Non-torus Mapsmentioning

confidence: 99%

“…The Levy decomposition of f (and equivalently of its biset) is its decomposition (as a graph of bisets) along the canonical Levy obstruction. It was proven in [20,Main Theorem II] that every Levy-free map that is doubly covered by a torus endomorphism is in tGTor/2u. Combined with Theorem A, this implies Corollary B.…”

Section: Introductionmentioning

confidence: 99%

Algorithmic aspects of branched coverings IV/V. Expanding maps

Bartholdi

Dudko

2018

Trans. Amer. Math. Soc.

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“…So uniqueness fails, and moreover, there is a non-pinching obstruction. For Thurston maps of type (2, 2, 2, 2) with additional marked points, pinching and non-pinching obstructions are characterized by Selinger-Yampolsky [49,50].…”

Section: Obstructions and The Thurston Theoremsmentioning

confidence: 99%

“…Sketch of the proof: 1. The lift to an affine map is explained in [14,23,34,50,6], see also [27]. In the flexible Lattès case, the choice of lattice is arbitrary, and in the case of non-real eigenvalues, the lattice can be chosen such that the real-affine map is holomorphic.…”

Section: Obstructions and The Thurston Theoremsmentioning

confidence: 99%

The Thurston Algorithm for quadratic matings

Jung

2017

Preprint

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Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless the orbifold of f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of postcritical rayequivalence classes for the modified mating, but it is enough to implement the pullback map for the formal mating. The proof combines the Selinger extension to augmented Teichmüller space with local estimates. Moreover, the Thurston Algorithm is implemented by pulling back a path in moduli space. This approach is due to Bartholdi-Nekrashevych in relation to one-dimensional moduli space maps, and to Buff-Chéritat for slow mating.Here it is shown that slow mating is equivalent to the Thurston Algorithm for the formal mating. An initialization of the path is obtained for repellingpreperiodic captures as well, which provide an alternative construction of matings.

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“…As a consequence, we deduce that Pilgrim's canonical obstruction is the union of the Levy obstruction (the multicurve along which S 2 is pinched to produce the Levy decomposition) and the rational obstruction (the multicurve along which the expanding maps of the Levy decomposition should be further pinched to give rational maps). Selinger and Yampolsky show in [34,Main Theorem I] that the canonical obstruction is computable. 0.4.…”

Section: Introductionmentioning

confidence: 99%

Algorithmic aspects of branched coverings

Bartholdi¹,

Dudko²

2017

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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This is a survey, and a condensed version, 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. Contents

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Constructive Geometrization of Thurston Maps and Decidability of Thurston Equivalence

Cited by 9 publications

References 24 publications

Algorithmic aspects of branched coverings IV/V. Expanding maps

Algorithmic aspects of branched coverings IV/V. Expanding maps

The Thurston Algorithm for quadratic matings

Algorithmic aspects of branched coverings

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