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73
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
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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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
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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Cited by 2 (2 self)
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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
Finite Subdivision Rules
 Conform. Geom. Dyn
, 2001
"... . We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively sub ..."
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Cited by 33 (8 self)
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. We introduce and study finite subdivision rules. A finite subdivision rule is a finite list of instructions which determines a subdivision of a given planar tiling. Given a finite subdivision rule and a planar tiling associated to it, we obtain an infinite sequence of tilings by recursively
Expansion complexes for finite subdivision rules
 I, Conform. Geom. Dyn
"... Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an ..."
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Cited by 12 (5 self)
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Abstract. This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a onetile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has
The Simplest Subdivision Scheme for Smoothing Polyhedra
, 1997
"... Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neighboring edgemidpoints. This refinement rule yields a C 1 surface and the surface has a piecewise quadratic parametrization except at finite number of isolated points. We analyze and improve the constru ..."
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Cited by 90 (7 self)
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Given a polyhedron, construct a new polyhedron by connecting every edgemidpoint to its four neighboring edgemidpoints. This refinement rule yields a C 1 surface and the surface has a piecewise quadratic parametrization except at finite number of isolated points. We analyze and improve
MODULUS OF UNBOUNDED VALENCE SUBDIVISION RULES
"... Abstract. Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings as ex ..."
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Abstract. Cannon, Floyd and Parry have studied the modulus of finite subdivision rules extensively. We investigate the properties of the modulus of subdivision rules with linear and exponential growth at every vertex, using barycentric subdivision and a subdivision rule for the Borromean rings
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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Cited by 5 (2 self)
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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
A finite subdivision rule for the ndimensional torus. Geometriae Dedicata
, 2012
"... Abstract. Cannon, Floyd, and Parry have studied subdivisions of the 2sphere extensively, especially those corresponding to 3manifolds, in an attempt to prove Cannon’s conjecture. There has been a recent interest in generalizing some of their tools, such as extremal length, to higher dimensions. We ..."
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Cited by 1 (1 self)
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. We define finite subdivision rules of dimension n, and find an n−1dimensional finite subdivision rule for the ndimensional torus, using a wellknown simplicial decomposition of the hypercube. 1.
Reversing Subdivision Rules: Local Linear Conditions and Observations on Inner Products
 Journal of Computational and Applied Mathematics
, 1999
"... In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple an ..."
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Cited by 26 (16 self)
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In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finitedimensional Cartesian vector space. We produced simple
Results 1  10
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73