Mixed Tiling Systems

Abstract

This thesis extends the theory of tiling iterated function systems developed in [BV18] and connects the theory to other areas of the tiling literature. The rst two chapters provide background material, introduce tiling iterated function systems, and discuss properties of the tilings generated from these systems. Simple examples are showcased and repeatedly referenced throughout the thesis. The third chapter links the symbolic tiling theory to Anderson and Putnam tiling theory [AP98] and the fourth chapter connects the theory to the work of Bandt on neighbour graphs [BM18]. An extension to the neighbour graph theory is proposed which allows the application of these techniques to a wider range of tiling iterated function systems. The nal and title chapter of this thesis extends the symbolic tiling theory to mixed tiling systems. The notation and general framework for creating tilings from a family of tiling iterated function systems is presented. The examples considered in the mixed setting cover one-dimensional tilings, tilings with statistical circular symmetry, and tilings made from tiles with no interior. It is explained how these ideas are related to V-variable and superfractal theory. Show more