Conformal Tilings Ii: Local Isomorphism, Hierarchy, And Conformal Type

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 Mobius transformations, motivating higher level hierarchies that will emerge in the next paper of this series.