Expansion Complexes for Finite Subdivision Rules II

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


This paper gives applications of earlier work of the authors on the use of expansion complexes for studying conformality of finite subdivision rules. The first application is that a one-tile rotationally invariant finite subdivision rule (with bounded valence and mesh approaching 0) has an invariant conformal structure, and hence is conformal. The paper next considers one-tile single valence finite subdivision rules. It is shown that an expansion map for such a finite subdivision rule can be conjugated to a linear map, and that the finite subdivision rule is conformal exactly when this linear map is either a dilation or has eigenvalues that are not real. Finally, an example is given of an irreducible finite subdivision rule that has a parabolic expansion complex and a hyperbolic expansion complex.

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