Combinations of Complex Dynamical Systems

Pilgrim, Kevin M.

doi:10.1007/b14147

Cited by 39 publications

(51 citation statements)

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“…The whole action, therefore, can be characterized by studying cycles of components. For each component C there are three cases (compare [15,Canonical Decomposition Theorem]): the composition F := F C of all coverings in the cycle (that is the first-return map of C) is one of the following: -a homeomorphism, -a Thurston map with parabolic orbifold, -a Thurston map with hyperbolic orbifold.…”

Section: Pilgrim's Conjecturementioning

confidence: 99%

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Thurston’s pullback map on the augmented Teichmüller space and applications

Selinger

2011

Invent. math.

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Let f be a postcritically finite branched self-cover of a 2-dimensional topological sphere. Such a map induces an analytic self-map σ f of a finite-dimensional Teichmüller space. We prove that this map extends continuously to the augmented Teichmüller space and give an explicit construction for this extension. This allows us to characterize the dynamics of Thurston's pullback map near invariant strata of the boundary of the augmented Teichmüller space. The resulting classification of invariant boundary strata is used to prove a conjecture by Pilgrim and to infer further properties of Thurston's pullback map. Our approach also yields new proofs of Thurston's theorem and Pilgrim's Canonical Obstruction theorem.

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Section: Pilgrim's Conjecturementioning

confidence: 99%

“…10 an application of our approach is given: we prove a conjecture by Kevin Pilgrim [15] (see Theorem 10.3). …”

Section: Introductionmentioning

confidence: 98%

Thurston’s pullback map on the augmented Teichmüller space and applications

Selinger

2011

Invent. math.

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“…Tuning by D (θ) corresponds to the classical ("formal") tuning procedure, as described, for example, in [Mil89,Pil03]. Consequently, the group K (11, D (1/3)) is generated by…”

Section: Figure 9 External Raysmentioning

confidence: 99%

Symbolic dynamics and self-similar groups

Nekrashevych¹

2008

Holomorphic Dynamics and Renormalization

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Abstract. Self-similar groups and permutational bimodules are used to study combinatorics and symbolic dynamics of expanding self-coverings. We describe functors between the category of contracting self-similar groups and the category of expanding self-coverings (with appropriate morphisms). These functors transform some questions in dynamical systems to questions in algebra. As examples we show how some plane filling curves (in particular the original Peano curve) can be interpreted in terms of embeddings of self-similar groups.

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“…These came up in Thurston's topological characterization of critically finite rational maps. For further information, see the Thurston notes [11], the Douady-Hubbard paper [6], the Pilgrim paper [8], or the Pilgrim monograph [9].…”

Section: Critically Finite Mapsmentioning

confidence: 99%

Constructing subdivision rules from rational maps

Cannon

Floyd

Parry

2007

Conform. Geom. Dyn.

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Abstract. This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f •n is the subdivision map of a finite subdivision rule. This enables one to give essentially combinatorial models for the dynamics of such iterates.We are interested here in connections between finite subdivision rules and rational maps. Finite subdivision rules arose out of our attempt to resolve Cannon's Conjecture: If G is a Gromov-hyperbolic group whose space at infinity is a 2-sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3-space. Cannon's Conjecture can be reduced (see, for example, the Cannon-Swenson paper [5]) to a conjecture about (combinatorial) conformality for the action of such a group G on its space at infinity, and finite subdivision rules were developed to give models for the action of such a Gromov-hyperbolic group on the 2-sphere at infinity.There is also a connection between finite subdivision rules and rational maps. 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 critically finite branched map of this 2-sphere. In joint work [3] with Kenyon we consider these subdivision maps under the additional hypotheses that R has bounded valence (this is equivalent to the subdivision map not having periodic critical points) and mesh approaching 0. In [3, Theorem 3.1] we show that if R is conformal (in the combinatorial sense), then the subdivision map σ R is equivalent to a rational map. The converse follows from [4, Theorem 4.7]. While a finite subdivision rule is defined in terms of a map from the subdivision complex to itself, it is basically a combinatorial procedure for recursively subdividing polygonal 2-complexes. By establishing the equivalence between a rational map and a subdivision map, one is giving an essentially combinatorial description of the action of the rational map.In this paper we consider the problem of when a rational map f is equivalent to the subdivision map of a finite subdivision rule. Since a subdivision complex has only finitely many vertices, such a rational map must be critically finite. We specialize here to the case that f has no periodic critical points. Our main theorem,

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Combinations of Complex Dynamical Systems

Cited by 39 publications

References 0 publications

Thurston’s pullback map on the augmented Teichmüller space and applications

Thurston’s pullback map on the augmented Teichmüller space and applications

Symbolic dynamics and self-similar groups

Constructing subdivision rules from rational maps

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