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Conformal Geometry and Dynamics

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Conformal tilings II: Local isomorphism, hierarchy, and conformal type

Authors: Philip L. Bowers and Kenneth Stephenson
Journal: Conform. Geom. Dyn. 23 (2019), 52-104
MSC (2010): Primary 52C23, 52C26; Secondary 52C45, 68U05
Published electronically: April 26, 2019
MathSciNet review: 3943256
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Abstract: This is the second in a series of papers on conformal tilings. The overriding themes here are local isomorphisms, hierarchical structures, and the conformal “type” problem. Conformal tilings were introduced by the authors in 1997 with a conformally regular pentagonal tiling of the plane. This and even more intricate hierarchical patterns arise when tilings are repeatedly subdivided. Deploying a notion of expansion complexes, we build two-way infinite combinatorial hierarchies and then study the associated conformal tilings. For certain subdivision rules the combinatorial hierarchical properties are faithfully mirrored in their concrete conformal realizations. Examples illustrate the theory throughout the paper. In particular, we study parabolic conformal hierarchies that display periodicities realized by Möbius transformations, motivating higher level hierarchies that will emerge in the next paper of this series.

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Additional Information

Philip L. Bowers
Affiliation: Department of Mathematics, The Florida State University, Tallahassee, Florida 32306
MR Author ID: 40455

Kenneth Stephenson
Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
MR Author ID: 216579

Received by editor(s): November 22, 2017
Received by editor(s) in revised form: August 7, 2018
Published electronically: April 26, 2019
Additional Notes: The second author gratefully acknowledges support of a Simons Foundation Collaboration Grant
Article copyright: © Copyright 2019 American Mathematical Society