TOPOLOGICAL SPACES AND DYNAMICAL SYSTEMS 13

Definition 2.4 If X is a subspace of Y, both topological spaces, and

A C X, then we write IntxA to mean the interior of A in the subspace

topology of X.

Likewise we write dxA for the boundary of A taken in the sub-

space topology of X.

Lemma 2.5 a/Ifu G NS, then (Q + u)nIntK = Int(Q+u}nE±((Q +

u) n K) and (Q + u) n ^ - L K = 9(Q+u)ni^ ((Q + w)flif).

b/IfueNS, then((Q + u)nE±)\NS = d(Q+u)nE±((Q + u)n

K)+7T±(ZN).

Proof a/ To show both facts, it is enough to show that (dE±K) H

(Q + u) has no interior as a subspace of (Q + u) n

E1-.

Suppose otherwise and that U is an open subset of dK D (Q + u)

in (Q + u)nE±. By the density of ^(ZN) in Q n ^ , we find i; G ZN

such that wEtZ + Tr

1

^). But this implies that u G dK +

7r-L(^)

and

so u 0 iVS - a contradiction.

b/ By defintion the left-hand side of the equation to be proved

is equal to (dE±K +

T T ^ Z ^ ) )

Ci(Q + u) which equals (dE±Kn (Q +

u)) +7r±(ZN) since 7rJ-(ZJV) is dense in Qn E±. By part a/ therefore

we obtain the right-hand side of the equation. •

Condition 2.6 We exclude immediately the case (Q + u)nIntK = 0

since, when u G NS, this is equivalent to Pu = 0.

Examples 2.7 We note the parameters of two well-studied examples,

both with canonical acceptance domain (2.3).

The octagonal tiling [Soc] has N = 4 and d = 2, where i? is a

vector subspace of

R4

invariant under the action of the linear map

which maps orthonormal basis vectors e\ H- e2, ^2 ^- e3, e3 i— e4,

e4 H-• —ei. Its orthocomplement, E1"1, is the other invariant subspace.

Here Q = R4 and so many of the distinctions made in subsequent

sections are irrelevant to this example.

The Penrose tiling [Pe] [dBl] has N = 5 and d = 2 (although

we note that there is an elegant construction using the root lattice of

A4 in

R4

[BJKS]). The linear map which maps e^ H- e^+i (indexed

modulo 5) has two 2 dimensional and one 1 dimensional invariant

subspaces. Of the first two subspaces, one is ^hosen &&JE and the

other we name V. Then in fact Q = E © V © A, where A = \{e\ +

^2 + 63 + e4 + es)Z, and Q is therefore a proper subset of

R5,

a fact

which allows the construction of generalized Penrose tilings using a

parameter u G NS \ Q.