TOPOLOGICAL SPACES AND DYNAMICAL SYSTEMS 15 and of full dimension. The projection cj) restricts to an orthogonal projection U1- U1- n F and, by construction, U1- D F n L = 0. Therefore Proposition 2.15 of [Se] applies to show that (j){L) is dense in U1- fl F and that \ is 1-1 on L. However j(L) C (j)(ZN) and so, by the characterisation of The- orem 2.2, we deduce that U1- n F C V. However, since U1- D V, we have U±nF = V. We have U n ZN = F n ZN and (^{U1- n Z^ ) = 0(Z N ) n y automatically. Therefore ((/)(ZN) HV) + (Fn ZN) = ^((t/ ^ n ZN) + (C7 n ZN)). As proved above, this latter set is the image of a finite index subgroup of the domain, Z ^ , and therefore it is a finite index subgroup of the image (f)(ZN) as required. The remaining properties follow quickly from the details above. Definition 2.10 Let A = n ZN and A = U n TT±(ZN) where U is the real vector space generated by A. Note that the discrete group A defined here is not the real vector space A (2?) defined in [Le], but it is a cocompact sublattice and so the dimensions are equal. Corollar y 2.11 With the notation of Theorem 2.8 and taking 0 = TT^, TT±(ZN) = V © A and Q = E 0 V 0 A are orthogonal direct sums. Moreover, A is a subgroup of A with finite index. E x a m p l e 2.12 For example the octagonal tiling has A = 0 and the Penrose tiling has A = (ei + e2 + es + e^ + e$)Z, a subgroup of index 5 in A. And finally a general result about isometric extensions of dynamical systems. Definition 2.13 Suppose that p: (X,G) {Y,G) is a factor map of topological dynamical systems with group, G, action. If every fibre p~1(y) has the same finite cardinality, n, then we say that (X, G) is an n-to-1 extension. The structure of such extensions, a special case of isometric exten- sions, is well-known [F].
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