## Expansion Complexes for Finite Subdivision Rules II

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

### Abstract

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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