An algebraic formulation of Thurston’s characterization of rational functions

Pilgrim, Kevin M.

doi:10.5802/afst.1361

Cited by 16 publications

(31 citation statements)

References 18 publications

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“…as in [8]. For NET maps, computation of the slope function σ f together with the mapping degree and number of preimages amounts to computation of φ f on powers of Dehn twists.…”

Section: Dehn Twistsmentioning

confidence: 99%

Nearly Euclidean Thurston maps

Cannon

Floyd

Parry

et al. 2012

Conform. Geom. Dyn.

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Abstract. We take an in-depth look at Thurston's combinatorial characterization of rational functions for a particular class of maps we call nearly Euclidean Thurston maps. These are orientation-preserving branched maps f : S 2 → S 2 whose local degree at every critical point is 2 and which have exactly four postcritical points. These maps are simple enough to be tractable, but are complicated enough to have interesting dynamics.

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“…as in [8]. For NET maps, computation of the slope function σ f together with the mapping degree and number of preimages amounts to computation of φ f on powers of Dehn twists.…”

Section: Dehn Twistsmentioning

confidence: 99%

Nearly Euclidean Thurston maps

Cannon

Floyd

Parry

et al. 2012

Conform. Geom. Dyn.

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“…This correspondence is crucial for enumerating the obstructed Q(4) * maps in §5, as well as computing the pullback on curves in §6 (see e.g. [3,21,12]). For any member of Q(4) * , the correspondence on moduli space restricts to a quadratic rational map.…”

Section: Ramification Portraits and Maps On Moduli Space In Q(4) *mentioning

confidence: 99%

“…Pilgrim has conjectured that if f is a rational map with hyperbolic orbifold, then µ f has a finite global attractor. The program NETmap [18] has provided a great deal of evidence in favor of this conjecture, and a number of results have been proven in special cases [21,12,10,7].…”

Section: Pullback On Essential Curvesmentioning

confidence: 99%

Quadratic Thurston maps with few postcritical points

Kelsey

Lodge

2018

Geom Dedicata

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We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on moduli space can be realized by quadratic rational maps with fewer than four postcritical points.Holomorphic dynamics in one complex variable is largely concerned with the study of rational maps as dynamical systems, along with their parameter spaces. Postcritically finite rational maps have attracted much attention due to their relative simplicity and their structural significance in parameter space. Furthermore, W. Thurston has given a powerful characterization and rigidity theorem for postcritically finite rational maps up to combinatorial equivalence (see Theorem 1.1). With this tool at our disposal, we are free to consider rational maps in the more flexible topological category.

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“…Restricted virtual homomorphism and other definitions. Here, we briefly comment on the difference between the definition of virtual homomorphism given here and that of the virtual endomorphism given in [Pil3].…”

Section: No Dynamicsmentioning

confidence: 99%

Pullback invariants of Thurston maps

Koch

Pilgrim

Selinger

2015

Trans. Amer. Math. Soc.

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Associated to a Thurston map f : S 2 → S 2 with postcritical set P are several different invariants obtained via pullback: a relation S P f ←− S P on the set S P of free homotopy classes of curves in S 2 \ P , a linear operator λ f : R[S P ] → R[S P ] on the free R-module generated by S P , a virtual endomorphism φ f : PMod(S 2 , P ) PMod(S 2 , P ) on the pure mapping class group, an analytic self-map σ f : T (S 2 , P ) → T (S 2 , P ) of an associated Teichmüller space, and an analytic self-correspondence X • Y −1 : M(S 2 , P ) ⇒ M(S 2 , P ) of an associated moduli space. Viewing these associated maps as invariants of f , we investigate relationships between their properties.

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An algebraic formulation of Thurston’s characterization of rational functions

Cited by 16 publications

References 18 publications

Nearly Euclidean Thurston maps

Nearly Euclidean Thurston maps

Quadratic Thurston maps with few postcritical points

Pullback invariants of Thurston maps

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