Tiling systems and 2-graphs associated to textile systems

This thesis concerns dynamical systems called tiling systems, and combinatorial structures called 2-graphs associated to textile systems. Textile systems were developed as a means of defining and studying two dimensional dynamical systems called shift spaces. We show how to construct a 2-graph from a textile system when the system has what we call unique path lifting, and investigate the relationship between the properties of the 2-graph and the dynamical properties of the tiling system. Our motivation is to link the properties of a 2-graph which appear in hypotheses of theorems about C*-algebras to key properties of dynamical systems. Specifically we have investigated the textile systems for higher block codings of tiling systems. We show that the 2-dimensional dynamical system associated to the 2-graph constructed from a textile system is conjugate to the tiling systems corresponding to the textile system. We investigate the periodicity of tilings arising from textile systems and give sufficient conditions to guarantee that there exist some tilings which are periodic. We also use ideas developed for 2-graphs to provide checkable conditions on a textile system under which the associated shift space is topologically transitive. We study the relationship between topological properties of the shift space of a textile system and the connectivity of the associated 2-graph. We also investigate the entropy of the tiling system, and show that it is zero when the textile system gives rise to a 2-graph. We construct examples of textile systems whose associated shift spaces have arbitrarily small nonzero entropy.