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 =

E±

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].