Elastic Graphs

Thurston, Dylan P.

doi:10.1017/fms.2019.4

Cited by 3 publications

(7 citation statements)

References 32 publications

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“…Given any , a ribbon elastic graph G can be thickened into a conformal surface with boundary by replacing each edge e of G by a rectangle of size and gluing the rectangles at the vertices by using the given ribbon structure. There are then inequalities relating and , to within a multiplicative factor as shown by the second author [57, Props. 4.8 and 4.9].…”

Section: Examplesmentioning

confidence: 99%

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From curves to currents

Martínez-Granado

Thurston

2021

Forum of Mathematics, Sigma

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Many natural real-valued functions of closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of length with respect to a fixed hyperbolic metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve function that guarantees a continuous extension to geodesic currents. The main condition of our criterion is the smoothing property, which has played a role in the study of systoles of translation lengths for Anosov representations. It is easy to see that our criterion is satisfied for almost all known examples of continuous functions on geodesic currents, such as nonpositively curved lengths or stable lengths for surface groups, while also applying to new examples like extremal length. We use this extension to obtain a new curve counting result for extremal length.

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Section: Examplesmentioning

confidence: 99%

“…There is a parallel notion of extremal length with respect to elastic graphs, as introduced by the second author [56], just as there is for ordinary lengths (Subsection 4.4).…”

Section: Extremal Length With Respect To Elastic Graphsmentioning

confidence: 99%

From curves to currents

Martínez-Granado

Thurston

2021

Forum of Mathematics, Sigma

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“…In recent work, Dylan Thurston discovered a positive characterization for rationality ( [7,8]). Central to the theory is an elastic graph spine G f in CP 1 − C f the existence of which is necessary and sufficient for the self-cover to be conjugate to a rational map.…”

Section: Rational Maps On Cp 1 As Branched Self-coversmentioning

confidence: 99%

Critically-Finite Dynamics on the Icosahedron

Crass

2020

Symmetry

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A recent effort used two rational maps on the Riemann sphere to produce polyhedral structures with properties exemplified by a soccer ball. A key feature of these maps is their respect for the rotational symmetries of the icosahedron. The present article shows how to build such “dynamical polyhedra” for other icosahedral maps. First, algebra associated with the icosahedron determines a special family of maps with 60 periodic critical points. The topological behavior of each map is then worked out and results in a geometric algorithm out of which emerges a system of edges—the dynamical polyhedron—in natural correspondence to a map’s topology. It does so in a procedure that is more robust than the earlier implementation. The descriptions of the maps’ geometric behavior fall into combinatorial classes the presentation of which concludes the paper.

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“…Asymptotic q-conformal energies. A virtual endomorphism of graphs pπ, φq has, for each q P r1, 8s, an associated asymptotic q-conformal graph energy E q pπ, φq, introduced by the second author [Thu19]. We summarize some key points; see §4 or the references for more.…”

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confidence: 99%

“…Energies of graph maps. ( §4) When equipped with natural length metrics, the maps φ n :" φ n 0 : G n Ñ G 0 have, for each q P r1, 8s, a q-conformal energy E q q rφ n s in the sense introduced by the second author in [Thu19]. The growth rate of this energy as n tends to infinity, namely E q rπ, φs :" lim E q q rφ n s 1{n , is called the asymptotic q-conformal energy, depends only on the homotopy class rπ, φs, and is non-increasing in q (Proposition 4.11).…”

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confidence: 99%

Ahlfors-regular conformal dimension and energies of graph maps

Pilgrim¹,

Thurston²

2021

Preprint

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For a hyperbolic rational map f with connected Julia set, we give upper and lower bounds on the Ahlfors-regular conformal dimension of its Julia set J f from a family of energies of associated graph maps. Concretely, the dynamics of f is faithfully encoded by a pair of maps π, φ : G 1 Ñ G 0 between finite graphs that satisfies a natural expanding condition. Associated to this combinatorial data, for each q ě 1, is a numerical invariant E q rπ, φs, its asymptotic q-conformal energy. We show that the Ahlfors-regular conformal dimension of J f is contained in the interval where E q " 1.Among other applications, we give two families of quartic rational maps with Ahlforsregular conformal dimension approaching 1 and 2, respectively.

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Elastic Graphs

Cited by 3 publications

References 32 publications

From curves to currents

From curves to currents

Critically-Finite Dynamics on the Icosahedron

Ahlfors-regular conformal dimension and energies of graph maps

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