14 1. BASIC NOTIONS
2(a) σ(a) a
Figure 1.11. There are two interior tiles of type a in
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
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
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.