A primitive, aperiodic substitution on d letters has at most d 2 asymptotic orbits; this bound is sharp. Since asymptotic arc components in tiling spaces associated with substitutions are in 1–1 correspondence with asymptotic words, this provides a bound for those as well.
We consider tiling dynamical systems and topological conjugacies between them. We prove that the criterion of being finite type is invariant under topological conju-gacy. For substitution tiling systems under rather general conditions, including the Penrose and pinwheel systems, we show that substitutions are invertible and that conjugacies are generalized… (More)
We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hyperbolic plane. Our goal is to understand qualitative… (More)