1.2. TILING SPACES 7 than 1/4, since translating both tilings a small distance to the left will cause them to agree on a slightly larger ball.) The first and third have distance 1/4, since they agree everywhere, up to a translation by 1/4. The second and third have distance at most 8/31, since after translating the first to the right by 1/8 4/31 and the second to the left by 1/8, they agree on a ball of radius 31/8 around the origin. Definition. The orbit of a tiling T is the set O(T ) = {T − x|x ∈ Rd} of translates of T . Definition. A tiling space is a set of tilings that is (1) closed under translation, and (2) complete in the tiling metric. That is, if T ∈ Ω then T − x ∈ Ω, and every Cauchy sequence of tilings in Ω has a limit in Ω. Definition. The hull ΩT of a tiling T is the closure of O(T ). This is sometimes called the “orbit closure” of T . The hull ΩT should be viewed as the set of tilings that locally look like translates of T . In particular, a tiling T is in ΩT if and only every patch of T is found in a translate of T . (Equivalently, a translate of every patch of T is found in T .) To see this, suppose that P is a patch in T , located somewhere in Br for some r 0. If T is in the hull of T , then there are translates of T that agree with T on arbitrarily large balls around the origin, hence that contain P . Conversely, let Pr = [Br] in T . If every patch of T is found in a translate of T , then there exist translates Tr of T that contain Pr. But then T = limr→∞ Tr is in the orbit closure of T . Examples of Hulls. To build our intuition, let’s look at the hulls of some simple 1-dimensional tilings. Let T0 be a periodic tiling with just one kind of tile: a white tile of length one. Since T0 − 1 = T0, the orbit of T is topologically the circle R/Z. This is already a complete metric space, so ΩT 0 is just a circle. Next consider the “one black tile” tiling T1. Every tiling with one black tile is in O(T1), and hence in ΩT 1 . However, tilings with no black tiles are also in ΩT 1 , since every patch of such a tiling can be found in T1, both sufficiently far to the right of the origin and sufficiently far to the left. ΩT 1 consists of two path-components: the circle ΩT 0 , and the line O(T1), with both ends of the line asymptotically approaching the circle, as in Figure 1.9. Exercise 1.1. What is ΩT when T is the half-and-half tiling? How many path components does it have? Exercise 1.2. Let T be a 1-dimensional tiling, with the color of the tiles (black or white) decided by fair and independent coin flips. What is ΩT ? Your answer could in principle depend on T , since it’s conceivable that your coin flips would always yield heads (T0) or yield tails exactly once
Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.