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

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- by Philip L. Bowers and Kenneth Stephenson PDF
- Conform. Geom. Dyn.
**23**(2019), 52-104 Request permission

## 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.## References

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## Additional Information

**Philip L. Bowers**- Affiliation: Department of Mathematics, The Florida State University, Tallahassee, Florida 32306
- MR Author ID: 40455
- Email: bowers@math.fsu.edu
**Kenneth Stephenson**- Affiliation: Department of Mathematics, The University of Tennessee, Knoxville, Tennessee 37996
- MR Author ID: 216579
- Email: kstephe2@utk.edu
- 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
- © Copyright 2019 American Mathematical Society
- Journal: Conform. Geom. Dyn.
**23**(2019), 52-104 - MSC (2010): Primary 52C23, 52C26; Secondary 52C45, 68U05
- DOI: https://doi.org/10.1090/ecgd/333
- MathSciNet review: 3943256