2 1. THE RUELLE ALGEBRA OF A RELATIVELY EXPANSIVE SYSTEM
tuple (X, d, Γ,f,E,E0) is a relatively expansive system. The most familiar examples
of expansive actions are actions of N, Z or Zn on compact metric spaces.
Example 1.2. Let G = (V, E) be a countable strongly connected, locally finite
directed graph with vertex set V , edge set E together with the maps i, t : E V ,
where i(e) is the initial and t(e) the terminal vertex of an edge e E. Then
XG = (xi)i∈Z
EZ
: t (xj) = i (xj+1) ∀j Z
is a locally compact subset of EZ and the shift σ acts as a homeomorphism of XG
in the standard way: σ(x)j = xj+1. Furthermore, the locally compact topology of
XG is given by a metric, called the Gurevich metric, cf. [Sch]. Unless G is finite
the shift is rarely expansive with respect to the Gurevich metric, but there is a
natural class of graphs for which the shift acts expansively on a canonical dense
subset: Assume that G only has finitely many pairwise edge-disjoint double paths,
as defined by Schraudner in [Sch]. As shown by Schraudner in Theorem 3.4 of [Sch]
there is then a constant c 0 such that supn∈Z d
(σn(x),σn(y))
c whenever x
and y are different points in XG and at least one of them is doubly transitive,
meaning that both the forward and the backward orbit is dense in XG. Note that
the doubly transitive points are dense in XG since G is strongly connected. Hence,
if we follow [Sch] and let DT (XG) denote the doubly transitive points of XG, the
tuple (XG,d, Z,σ,XG,DT (XG)) will be a relatively expansive system.
1.2. The ´ etale equivalence relation of local conjugacy
We recall the definition of an ´ etale equivalence relation, cf. [Re1], [GPS].
Let X be a set and R X × X an equivalence relation. We say that R is
a topological equivalence relation when R is equipped with a topology (possibly
different from the topology inherited from X × X) such that the inversion R
(x, y) (y, x) R is a homeomorphism and the composition
R(2)
((x, y), (y, z)) (x, z) R
is continuous, where the set of composable pairs
R(2)
= {((x, y), (u, v)) R × R : u = y}
has the relative topology inherited from R × R. In this setting we call r(x, y) = x
the range map and s(x, y) = y the source map.
Definition 1.3. Let X be a locally compact Hausdorff space and R X × X
a topological equivalence relation. R is an ´ etale equivalence relation when the range
map r : R X is a local homeomorphism in the sense that every element γ R has
an open neighborhood of γ such that r (Uγ) is open in X and r : r (Uγ )
is a homeomorphism.
We come now to the basic construction of the paper; the construction of an
´ etale equivalence relation from a relatively expansive system (X, d, S, f, E, E0).
Two elements x, y E are said to be locally conjugate, written x y, when
there are open neighborhoods U and V of x and y in E, and a homeomorphism
χ : U V such that χ(x) = y and
lim
s→∞
sup
z∈U
d (fs(z),fs (χ(z))) = 0.
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