14 1. BASIC NOTIONS (a) σ 2 σ(a) a Figure 1.11. There are two interior tiles of type a in σ2(a), but none in σ(a). Figure 1.12. A self-similar chair tiling take x in the interior of t2, which in turn means taking t2 in the interior of t1. Historically, substitution tilings were often studied by looking at self- similar tilings and then considering the hulls of these tilings. The following theorem, combined with the existence of self-similar tilings, shows that this approach yields exactly the same spaces as we have been considering. Theorem 1.5. If σ is a primitive substitution and T ∈ Ωσ, then O(T ) is dense in Ωσ, and Ωσ = ΩT . Sketch of proof. We must show that if T ∈ Ωσ, then every patch of T is found somewhere in T . To do this, we show that for each tile t and integer n ≥ 0, there exists a radius R such that every ball of radius R in T contains a copy of σn(t). For n = 0 this comes from the primitivity of σ. For large values of n it follows from the n = 0 result combined with the fact that σ is invertible on Ωσ. Exercise 1.9. Expand this sketch into a complete proof.

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