8 1. BASIC NOTIONS
Figure 1.9. The hull of “one black tile”
(T1). However, those are probability zero events. There is a space that ΩT
will equal with probability one. Find it.
Theorem 1.1. If T is a simple tiling, then ΩT is compact.
Proof. We must show that every sequence in ΩT has a convergent
subsequence. Since there are only a finite number of tile types, and there
are only a finite number of ways in which tiles can abut, for each r there
are only a finite number of possible patches [Br], up to translation by a
distance smaller than the diameter of the largest tile. Therefore, in any
sequence of tilings in ΩT , there is a subsequence that converges on Br. Now
apply the Cantor diagonalization trick. Of the subsequence that converges
on B1, pick a sub-subsequence that converges on B2, a subsequence of that
that converges on B3, and so on. Take the first element of the sequence
that converges on B1, the second element of the sequence that converges
on B2, etc. This sequence converges on every bounded set, and so forms a
Cauchy sequence in the tiling metric. Since the tiling space is complete, this
sequence has a limit in ΩT .
1.3. Equivalence
In topology, we are always classifying spaces up to equivalence. But
what does it mean to say that two tiling tiling spaces are equivalent? There
are several different notions, the weakest of which is homeomorphism.
Definition. A homeomorphism between (simple) tiling spaces is a con-
tinuous map f : ΩT ΩT that is 1-1 and onto. Since ΩT is compact,
f−1
is
automatically continuous, so this agrees with the usual topological definition
of homeomorphism.
Homeomorphisms preserve topology, but little else.
Definition. A factor map between tiling spaces is a map that commutes
with the action of the translation group. A topological conjugacy between
tiling spaces is a homeomorphism that is also a factor map.
Topological conjugacies preserve the structure of tiling spaces as dynam-
ical systems targets for the action of the translation group. Topological
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