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
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 then
T x Ω, and every Cauchy sequence of tilings in 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
ΩT0 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 ΩT1 . However, tilings with no black tiles are
also in ΩT1 , 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. ΩT1
consists of two path-components: the circle ΩT0 , 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
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