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)n i^ ((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 (d E ±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 P u = 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.
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