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1.4. CONTRUCTING INTERESTING TILINGS 11 Definition. If σ is a primitive substitution on an alphabet {a1, . . . , ak}, we call a bi-infinite word σ-admissible if each finite sub-word can be found in σn(a1) for some n ≥ 0. In such cases, let L = (L1, . . . , Lk) be a left-eigenvector of the substitu- tion matrix corresponding to λP F , and consider tiles with labels a1, . . . , ak and lengths L1, . . . , Lk. Exercise 1.3. Show that the length of the patch corresponding to σn(aj) is λn P F Lj. Exercise 1.4. Show that, as n → ∞, the fraction of letters of type ai in σn(aj) approaches Ri/ k Rk, and in particular is independent of j. Definition. Given a substitution σ, the sequence space Sσ is the set of all bi-infinite σ-admissible words and the tiling space Ωσ consists of all tilings by the tiles (a1, . . . , ak) such that the corresponding sequence of letters (a1, . . . , ak) forms a σ-admissible word. The substitution σ acts on Sσ by replacing each letter with a word. It acts on Ωσ by first stretching the tiling by a factor of λP F about the origin, and then replacing each stretched tile of type aj with tiles corresponding to the word σ(aj). Exercise 1.5. Show that for any integer k 0 and any substitution σ, Ωσ = Ω σk Definition. A patch of a tiling corresponding to σn(ai) is called a su- pertile of level n (or order n) and type i. Ωσ is the set of tilings with the property that every patch is found inside a supertile of some order. Theorem 1.3. [Mos, Sol] If σ is a primitive substitution and Ωσ con- tains at least one non-periodic tiling, then every tiling in Ωσ is non-periodic and σ : Ωσ → Ωσ is a homeomorphism. In particular, if T ∈ Ωσ, then there is a unique way to group the tiles of T into supertiles of order 1, such that the pattern of supertiles looks like a scaled-up version of a tiling in Ωσ. Examples. (1) Thue-Morse sequences have an alphabet of two letters, a and b, with the substitution σ(a) = ab, σ(b) = ba. Note that σ2(a) = abba be- gins with a and σ2(b) = baab ends with b. Let w = b.a, with the dot indicating the location of the origin. σ2(w) = baab.abba con- tains w in the center. Likewise, σ4(w) contains σ2(w), and gener- ally σn+2(w) contains σn(w). Taking limits we find a sequence u = . . . abbaabbabaab.abbabaabbaababba . . . with σ2(u) = u. We can make a tiling T out of u by thinking of each letter as a tile of length 1. Exercise 1.6. Consider the patch bbaababbabaababbaabbaba of a Thue-Morse tiling. Group the tiles into supertiles of level 1, 2, etc., as far as you can go. Note that near the edges, a supertile may only be partially in the patch.