Expanding Thurston Maps

Bonk, Mario; Meyer, Daniel

doi:10.1090/surv/225

Cited by 84 publications

(267 citation statements)

References 47 publications

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“…Concatenating γ with a subpath of η and applying (15) and (16) now gives a path γ ′ / ∈ Γ 0 joining ζ 1 and ζ 3 in Q such that…”

Section: Continuity Of Umentioning

confidence: 99%

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Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

156

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Abstract. We establish uniformization results for metric spaces that are homeomorphic to the Euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and sufficient condition for such spaces to be QC equivalent to the Euclidean plane, disk, or sphere. Moreover, we show that if such a QC parametrization exists, then the dilatation can be bounded by 2. As an application, we show that the Euclidean upper bound for measures of balls is a sufficient condition for the existence of a 2-QC parametrization. This result gives a new approach to the Bonk-Kleiner theorem on parametrizations of Ahlfors 2-regular spheres by quasisymmetric maps.

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“…Concatenating γ with a subpath of η and applying (15) and (16) now gives a path γ ′ / ∈ Γ 0 joining ζ 1 and ζ 3 in Q such that…”

Section: Continuity Of Umentioning

confidence: 99%

“…Also, the infinitesimal or analytic definitions of quasiconformality do not give a good theory in this case, and one has to concentrate on QS maps. See [54] and [27] for the basic properties of QS maps in metric spaces, and [15], [18], [41], [42] for results on QS parametrizations of fractal spaces.…”

mentioning

confidence: 99%

Uniformization of two-dimensional metric surfaces

Rajala

2016

Invent. math.

156

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“…7.3, we remarked that there are two analytic quantities associated to a virtual endomorphism of a graph: the asymptotic stretch factor SF and the asymptotic Lipschitz constant Lip. In the setting of Theorem 8, Bonk and Meyer [6] and Haïssinsky and Pilgrim [22] studied essentially this condition, in the opposite situation to the present paper: they considered the case when there are no branch points in any cycle in P. (For rational maps, this means that the Julia set is the whole sphere.) In our language, they showed more or less that for a branched self-cover f with no branch points in periodic cycles with associated virtual endomorphism φ, the stretch factor Lip[φ] < 1 iff f has a topological representative as a map on S 2 that is uniformly expanding.…”

Section: Comparison To Lipschitz Expansionmentioning

confidence: 95%

From rubber bands to rational maps: a research report

Thurston

2016

Res Math Sci

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This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. On one hand, this lets us tell when one rubber band network is looser than another and, on the other hand, tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.

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“…We provide a brief overview here, but refer the reader to [, Chapter 2] for details. Suppose

f : S^{2} \to S^{2}

is a branched covering map of the topological 2‐sphere

S^{2}

.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we consider Thurston maps defined by a subdivision rule on the sphere

S^{2}

. See for example the construction of the map

f_{1}

in Section 3; in greater generality this may be found in [, Chapter 12; ].…”

Section: Introductionmentioning

confidence: 99%

Exponential growth of some iterated monodromy groups

Hlushchanka

Meyer

2018

Proc. London Math. Soc.

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Abstract. Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have exponential growth. These groups arise from polynomials. We show exponential growth of the IMG of several non-polynomial maps. These include rational maps whose Julia set is the whole sphere, rational maps with Sierpiński carpet Julia set, and obstructed Thurston maps. Furthermore, we construct the first example of a non-renormalizable polynomial with a dendrite Julia set whose IMG has exponential growth.

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Expanding Thurston Maps

Cited by 84 publications

References 47 publications

Uniformization of two-dimensional metric surfaces

Uniformization of two-dimensional metric surfaces

From rubber bands to rational maps: a research report

Exponential growth of some iterated monodromy groups

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