## Finite Subdivision Rules

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J. W. Cannon, W. J. Floyd, and W. R. Parry

September 15, 1999

### Abstract

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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