Finite subdivision rules

Abstract:

We introduce and study finite subdivision rules. A finite subdivision rule $\mathcal {R}$ consists of a finite 2-dimensional CW complex $S_{\mathcal {R}}$, a subdivision $\mathcal {R}(S_{\mathcal {R}})$ of $S_{\mathcal {R}}$, and a continuous cellular map $\varphi _{\mathcal {R}}\colon \thinspace \mathcal {R}(S_{\mathcal {R}}) \to S_{\mathcal {R}}$ whose restriction to each open cell is a homeomorphism. If $\mathcal {R}$ is a finite subdivision rule, $X$ is a 2-dimensional CW complex, and $f\colon \thinspace X\to S_{\mathcal {R}}$ is a continuous cellular map whose restriction to each open cell is a homeomorphism, then we can recursively subdivide $X$ to obtain an infinite sequence of tilings. We wish to determine 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.