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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