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].
Previous Page Next Page