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Constructing rational maps from subdivision rules
, 2003
"... Suppose R is an orientationpreserving finite subdivision rule with an edge pairing. Then the subdivision map σR is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2sphere. If R has mesh approaching 0 and SR is a 2sphere, it is proved in Theorem 3.1 th ..."
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Cited by 14 (3 self)
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Suppose R is an orientationpreserving finite subdivision rule with an edge pairing. Then the subdivision map σR is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2sphere. If R has mesh approaching 0 and SR is a 2sphere, it is proved in Theorem 3.1 that if R is conformal then σR is realizable by a rational map. Furthermore, a general construction is given which, starting with a one tile rotationally invariant finite subdivision rule, produces a finite subdivision rule Q with an edge pairing such that σQ is realizable by a rational map.
Invariant Peano curves of expanding Thurston maps
, 2009
"... We consider Thurston maps, i.e., branched covering maps f: S² → S² that are postcritically finite. It is shown that a Thurston map f is expanding (in a suitable sense) if and only if some iterate F = f n is semiconjugate to z d: S 1 → S 1, where d = deg F. More precisely, for such an F we construc ..."
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Cited by 6 (5 self)
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We consider Thurston maps, i.e., branched covering maps f: S² → S² that are postcritically finite. It is shown that a Thurston map f is expanding (in a suitable sense) if and only if some iterate F = f n is semiconjugate to z d: S 1 → S 1, where d = deg F. More precisely, for such an F we construct a Peano curve γ: S 1 → S 2 (onto), such that F ◦ γ(z) = γ(z d) (for all z ∈ S 1).
CONSTRUCTING SUBDIVISION RULES FROM RATIONAL MAPS
"... 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 ..."
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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. 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 Gromovhyperbolic group whose space at infinity is a 2sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3space. Cannon’s Conjecture can be reduced (see, for example, the CannonSwenson 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 a Gromovhyperbolic group on the 2sphere at infinity. There is also a connection between finite subdivision rules and rational
SQUARING RECTANGLES FOR DUMBBELLS
"... Abstract. The theorem on squaring a rectangle (see Schramm [6] and CannonFloydParry [1]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectanglesquaring as an approximation to the classical Riemamnn mapping. 1. ..."
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Abstract. The theorem on squaring a rectangle (see Schramm [6] and CannonFloydParry [1]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectanglesquaring as an approximation to the classical Riemamnn mapping. 1.
SUBDIVISION RULES AND VIRTUAL ENDOMORPHISMS
"... Abstract. Suppose f: S 2 → S 2 is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting ..."
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Abstract. Suppose f: S 2 → S 2 is a postcritically finite branched covering without periodic branch points. If f is the subdivision map of a finite subdivision rule with mesh going to zero combinatorially, then the virtual endomorphism on the orbifold fundamental group associated to f is contracting. This is a first step in a program to clarify the relationships among various notions of expansion for noninvertible dynamical systems with branching behavior. Let T 2 = R 2 /Z 2 denote the real twodimensional torus, equipped with the Euclidean Riemannian metric ds 2 inherited from the usual metric on R 2, and suppose f: T 2 → T 2 is a continuous orientationpreserving covering map. It is wellknown that a necessary and sufficient condition for f to be homotopic to a covering map
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"... Let f: S2 → S2 be a branched covering of degree d ≥ 2 with branch set Bf. If the postcritical set ..."
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Let f: S2 → S2 be a branched covering of degree d ≥ 2 with branch set Bf. If the postcritical set
LATTÈS MAPS AND FINITE SUBDIVISION RULES
"... 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 ..."
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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 1skeleton 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 orientationpreserving finite subdivision rule such that the subdivision complex SR is a 2sphere, then the subdivision map σR is a postcritically finite branched map. Furthermore, R has bounded valence if and only if σR has no periodic critical points. In [1] and [4], BonkMeyer and CannonFloydParry 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 1skeleton of SR is a
Contents
, 2008
"... We continue the study of noninvertible topological dynamical systems with expanding behavior. We introduce the class of finite type systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbo ..."
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We continue the study of noninvertible topological dynamical systems with expanding behavior. We introduce the class of finite type systems which are characterized by the condition that, up to rescaling and uniformly bounded distortion, there are only finitely many iterates. We show that subhyperbolic rational maps and finite subdivision rules (in the sense of Cannon, Floyd, Kenyon, and Parry) with bounded valence and mesh going to zero are of finite type. In addition, we show that the limit dynamical system associated to a selfsimilar, contracting, recurrent, leveltransitive group action (in the sense of V. Nekrashevych) is of finite type. The proof makes essential use of an analog of the finiteness of cone types property enjoyed by hyperbolic groups.