14 1. BASIC NOTIONS

σ

2(a) 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.