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

This is the second in a series of papers on conformal tilings. The overriding themes of this paper are local isomorphisms, hierarchical structures, and the type problem in the context of conformally regular tilings, a class of tilings introduced first by the authors in 1997 with an example of a conformally regular pentagonal tiling of the plane [2]. We prove that when a conformal tiling has a combinatorial hierarchy for which the subdivision operator is expansive and conformal, then the tiling is parabolic and tiles the complex plane C. This is used to examine type across local isomorphism classes of tilings and to show that any conformal tiling of bounded degree that is locally isomorphic to a tiling obtained as an expansion complex of a shrinking and dihedrally symmetric subdivision operator with one polygonal type is parabolic.