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Constructing rational maps from subdivision rules. (English) Zbl 1077.37037

From the abstract: Suppose \(\mathcal{R}\) is an orientation-preserving finite subdivision rule with an edge pairing. Then the subdivision map \(\sigma _{\mathcal{R}}\) is either a homeomorphism, a covering of a torus, or a critically finite branched covering of a 2-sphere. If \(\mathcal{R}\) has mesh approaching 0 and \(S_{\mathcal{R}}\) is a 2-sphere, it is proved that if \(\mathcal{R}\) is conformal, then \(\sigma _{\mathcal{R}}\) 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 \(\mathcal{Q}\) with an edge pairing such that \(\sigma _{ \mathcal{Q}}\) is realizable by a rational map.

MSC:

37F20 Combinatorics and topology in relation with holomorphic dynamical systems
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
57M12 Low-dimensional topology of special (e.g., branched) coverings
20F67 Hyperbolic groups and nonpositively curved groups
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