Subdivision rules and virtual endomorphisms

Cannon, J. W.; Floyd, William J.; Parry, Walter; Pilgrim, Kevin M.

doi:10.1007/s10711-009-9352-7

Cited by 8 publications

(9 citation statements)

References 7 publications

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“…Proof. To prove this, we interpret condition 2 in terms of a directed graph G which was considered previously in [5]. The vertices of G are ordered pairs (e 1 , e 2 ), where e 1 and e 2 are disjoint edges of the tile type t of R. There exists a directed edge from the vertex (e 1 , e 2 ) to the vertex (e 3 , e 4 ) if and only if R(t) contains a tile s such that an edge of s with edge type e 3 is contained in e 1 and an edge of s with edge type e 4 is contained in e 2 .…”

Section: So R 1+τmentioning

confidence: 99%

Lattès maps and finite subdivision rules

Cannon

Floyd

Parry

2010

Conform. Geom. Dyn.

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Abstract. This paper is concerned with realizing Lattès maps as subdivision maps of finite subdivision rules. The main result is that the Lattès maps in all but finitely many analytic conjugacy classes can be realized as subdivision maps of finite subdivision rules with one tile type. An example is given of a Lattès map which is not the subdivision map of a finite subdivision rule with either i) two tile types and 1-skeleton of the subdivision complex a circle or ii) one tile type.This paper is concerned with realizing rational maps by subdivision maps of finite subdivision rules. If R is an orientation-preserving finite subdivision rule such that the subdivision complex S R is a 2-sphere, then the subdivision map σ R is a postcritically finite branched map. Furthermore, R has bounded valence if and only if σ R has no periodic critical points. In [1] and [4], Bonk-Meyer and CannonFloyd-Parry prove that if f is a postcritically finite rational map without periodic critical points, then every sufficiently large iterate of f is the subdivision map of a finite subdivision rule R with two tile types such that the 1-skeleton of S R is a circle. Since finite subdivision rules are essentially combinatorial objects, this gives good combinatorial models for these iterates. It is especially convenient to realize a postcritically finite map by the subdivision map of a finite subdivision rule with either a single tile type or with two tile types and 1-skeleton of the subdivision complex a circle.While passing to an iterate is not usually a serious obstacle, it would be preferable if one didn't need to do this. Suppose f is a postcritically finite rational map without periodic critical points. We consider the following questions.(1) Is f the subdivision map of a finite subdivision rule? (2) Is f the subdivision map of a finite subdivision rule with two tile types and 1-skeleton of the subdivision complex a circle? (3) Is f the subdivision map of a finite subdivision rule with one tile type? In this paper we consider these questions for Lattès maps. The main result of this paper is to answer question 3 in the affirmative for the maps in all but finitely many analytic conjugacy classes of Lattès maps. In addition we exhibit a Lattès map of degree 2 for which the answer to questions 2 and 3 is negative, although the answer to question 1 is positive. This paper has four sections. The first section develops the setting of Lattès maps. These results can be used to easily enumerate all Lattès maps of very small degree to test (not so easy) the above three questions for them.

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Section: So R 1+τmentioning

confidence: 99%

Lattès maps and finite subdivision rules

Cannon

Floyd

Parry

2010

Conform. Geom. Dyn.

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“…Combining these results with Theorem 6.5 implies that if R has bounded valence and the mesh of R approaches 0 combinatorially, then φ f is contracting. This is the main result of [11].…”

Section: Introductionmentioning

confidence: 68%

“…Proof. The proof follows the argument in the proof of [11,Theorem 6.1]. If the conditions (Esep), (VEsep), and (Vsep) are satisfied for n = kl 2 , then R is combinatorially expanding.…”

Section: Expansion Properties For Finite Subdivision Rulesmentioning

confidence: 82%

“…In [11,Theorem 6.1] we prove that, to verify the conditions (Esub) and (Esep) of (M0comb) for a finite subdivision rule R, it suffices to let n be the integer kl 2 , where k is the number of tiles in S R and l is the maximum number of edges of a tile type of R. The statement and proof easily adapt to give the following. Theorem 3.5.…”

Section: Expansion Properties For Finite Subdivision Rulesmentioning

confidence: 99%

“…Such Thurston maps are, a priori, special among general Thurston maps. Along the way, various notions of topological expansion and of combinatorial expansion were formulated for Thurston maps, and relationships among these notions were established [3,5,11,16]. In independent remarkable developments, it was shown that an expanding rational Thurston map and, more generally, an expanding Thurston map, has the property that some iterate is the subdivision map of an fsr [9,3].…”

Section: Introductionmentioning

confidence: 99%

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Expansion properties for finite subdivision rules I

2018

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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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Subdivision rules and virtual endomorphisms

Cited by 8 publications

References 7 publications

Lattès maps and finite subdivision rules

Lattès maps and finite subdivision rules

Expansion properties for finite subdivision rules I

Pullback invariants of Thurston maps

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