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Hyperbolic geometry
 In Flavors of geometry
, 1997
"... 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65 ..."
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3. Why Call it Hyperbolic Geometry? 63 4. Understanding the OneDimensional Case 65
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