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 → fγ ∈ 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