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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