Distribution of patches in tilings, tiling spaces and tiling dynamical systems (Natural extension of arithmetic algorithms and S-adic system)

Abstract

We introduce a basic theory of tilings, continuous hulls and tiling dynamical systems with detailed proofs. As a way to construct interesting tilings, tiling substitution is investigated. We put emphasis on two points: first, we introduce two topologies on the space of patches (and so the space of tilings) and describe the relation between them. Second, we discuss relations between properties of tilings and those of continuous hulls and tiling dynamical systems.