CHAPTER 1
The Ruelle algebra of a relatively expansive
system
1.1. Relatively expansive systems
We describe the input from dynamical systems which we need for the construc-
tion of the ´ etale equivalence relations and
C∗-algebras
we are going to study.
Let X be a topological space and d a metric for the topology of X. Let E X
be a subset of X equipped with a locally compact topology which is finer than the
topology inherited from X. That is, U E is open in E when U X is open in
X, but there may be open sets in E which are not of this form. Note that E is a
locally compact Hausdorff space since the topology inherited from X is Hausdorff.
Let S be a countable set and for each s S, let fs : X X be a continuous
map. Thus f = (fs)s∈S is simply a collection of continuous self-maps of X, indexed
by the set S. S may be a group or a semi-group, and s fs a homomorphism, but
this is not necessary for the basic construction we describe below. We will assume
that f is relatively expansive on E in the sense that there is a dense subset E0 of
E with the following two properties.1
1) E0 is asymptotically stable in the sense that when x E, y E0, and
lim
s→∞
d (fs(x),fs(y)) = 0,
then x E0.
2) f is locally expansive on E0 in the sense that for each x E there is an
open neighborhood Ux of x in E and a δx 0 such that
(1.1) z, y E0 Ux, d (fs(z),fs(y)) δx ∀s S z = y.
We call then the pair (Ux,δx) an expansive pair at x, and δx 0 is called a local
expansive constant at x. We say that E is an expansive region for the action f. The
tuple (X, d, S, f, E, E0) will be called a relatively expansive system in the following.
1.1.1. Examples.
Example 1.1. Let (X, d) be a locally compact metric space, End X the semi-
group of continuous maps from X to itself. Let Γ be a countable discrete semi-group.
An action of Γ on X is a semi-group homomorphism Γ γ End X. The
action is called expansive when there is a δ 0 such that
x, y X, sup
γ∈Γ
d (fγ(x),fγ(y)) δ x = y.
Then (X, d, Γ,f,X,X) is a relatively expansive system. More generally with E any
open or closed subset of X, and E0 any dense asymptotically stable subset of E, the
1Given a function G : S [0, ∞) we write lims→∞ G(s) = 0 to mean that for every 0
there is a finite set F S such that G(s) when s / F .
1
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