1.4. CONTRUCTING INTERESTING TILINGS 9 conjugacies preserve dynamical invariants such as the set of translation- invariant probability measures, the dynamical spectrum, and mixing prop- erties. The strongest notion of equivalence is Mutual Local Derivability, or MLD. [BSJ] Definition. Two tiling spaces are MLD if there is a topological con- jugacy between them that is defined locally. More precisely, there exists a radius R such that, whenever two tilings T1, T2 agree on a ball of radius R around x, then f(T1) and f(T2) agree on a ball of radius 1 around x. Since f commutes with translations, it is suﬃcient to check this at x = 0. That is, the properties of f(T ) near the point x are determined by the properties of T on some ball around x. The term MLD was originally defined for tilings, rather than for tiling spaces. Definition. If T and T are tilings, we say that T is locally derivable from T if, for some finite radius R, the properties of T at each point x are determined by the properties of T in a ball of radius R around x. Stated formally, if there exists a radius R such that, whenever points x and y have the property that [BR + x]T = [BR + y]T + y − x, then [B1 + x]T = [B1 + y]T + y − x. If T is locally derivable from T and T is locally derivable from T , then T and T are MLD. Each Penrose chicken tiling is MLD to a Penrose kite-and-dart tiling. The periodic “all white” tiling of R is locally derivable from “one black tile”, but “one black tile” is not locally derivable from “all white”. If the tilings T and T are MLD, then the tiling spaces ΩT and Ω T are automatically MLD. Just let f(T ) = T and extend this to the orbit of T by f(T + x) = T + x. The map f is uniformly continuous on O(T ). Given , pick δ ( −1 + R)−1. If T − x and T − y are δ-close, then the patterns in T around x and y agree (up to translation) out to distance δ−1, so the patterns of T around x and y agree (up to the same translations) up to distance δ−1 − R −1 , so f(T − x) = T − x and f(T − y) = T − y are -close. We can therefore extend the map f to all of ΩT by continuity. Conversely, if Ω and Ω are tiling spaces that are MLD with map f : Ω → Ω , and if T ∈ Ω, then the tilings T and f(T ) are MLD. For a long time, it was believed that topologically conjugate tiling spaces were automatically MLD. In 1999, however, two independent papers [Pet, RS] showed that this was false. In Chapter 3 we will see how changing the shapes and sizes of tiles can sometimes yield spaces that are topologically conjugate but not MLD. 1.4. Contructing interesting tilings There are three classes of tilings that come up repeatedly in tiling theory. First, there are substitution tilings. These are closely related to self-similar

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