A positive characterization of rational maps

Thurston, Dylan P.

doi:10.48550/arxiv.1612.04424

Cited by 5 publications

(5 citation statements)

References 10 publications

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“…In turn, this control over when one surface conformally embeds in another (plus a relation between extremal length on a graph and extremal length on its thickening) lets us prove a new characterization of when a topological branched self-cover of the sphere is equivalent to a rational map [Thu16b]. This gives a converse to an earlier characterization by W. Thurston [DH93].…”

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confidence: 85%

Elastic Graphs

Thurston

2019

Forum of Mathematics, Sigma

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An elastic graph is a graph with an elasticity associated to each edge. It may be viewed as a network made out of ideal rubber bands. If the rubber bands are stretched on a target space there is an elastic energy. We characterize when a homotopy class of maps from one elastic graph to another is loosening, i.e., decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs.

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confidence: 85%

Elastic Graphs

Thurston

2019

Forum of Mathematics, Sigma

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“…Given an unobstructed post-critically finite unicritical topological polynomial, one can use our algorithm to determine the combinatorics of the Hubbard tree for the polynomial in its Thurston class and then use the spider algorithm to find the coefficients of this polynomial. D. Thurston studies the case of post-critically finite branched covers of the sphere where each cycle of post-critical points contains a critical point [38]. He gives a positive characterization for such a map to be equivalent to a rational map.…”

Section: Comparisons To Prior Workmentioning

confidence: 99%

Recognizing topological polynomials by lifting trees

et al. 2022

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We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the canonical obstruction. Our approach is rooted in geometric group theory, using iteration on a simplicial complex of trees, and building on work of Nekrashevych. As one application of our methods, we resolve the polynomial case of Pilgrim's finite global attractor conjecture. We also give a new solution to Hubbard's twisted rabbit problem, and we state and solve several generalizations of Hubbard's problem where the number of post-critical points is arbitrarily large.

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“…-Dylan P. Thurston [55] gives a positive criterion for g to be equivalent to a rational map, at least if there is a periodic critical point: f −k is uniformly contracting on a graph, which forms a spine for C \ P . In [44], Kevin Pilgrim gives an algebraic characterization of obstructions by non-contraction of the virtual endomorphism of the pure mapping class group.…”

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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A positive characterization of rational maps

Cited by 5 publications

References 10 publications

Elastic Graphs

Elastic Graphs

Recognizing topological polynomials by lifting trees

The Thurston Algorithm for quadratic matings

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