Finite Subdivision Rules

J. W. Cannon, W. J. Floyd, and W. R. Parry
September 15, 1999


Given a finite subdivision rule and a tiling of a topological surface X which is compatible with it, one can recursively define a sequence of subtilings of the tiling of X. This paper is concerned with determining when this sequence of tilings is conformal in the sense of Cannon's combinatorial Riemann mapping theorem. In this setting, it is proved that the two axioms of conformality can be replaced by a single axiom which is implied by either of them, and that it suffices to check conformality for finitely many test annuli. Theorems are given which show how to exploit symmetry, and many examples are computed.

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