Preface
The crossed product construction is the classical way of associating a
C∗-algebra
to a dynamical system, and it is sometimes interpreted as a non-commutative sub-
stitute for the often ill behaved space of orbits of the action. This view of the crossed
product is most natural when the action is free, since the crossed product will other-
wise depend on more than the orbit equivalence relation. Nonetheless it makes good
sense to view the crossed product as an attempt to produce a non-commutative al-
gebra, a
C∗-algebra,
from the orbit equivalence defined by the action, in order to
capture important features of the orbit space which may be difficult or impossible
to handle from a topological point of view.
In recent years there has been a growing interest in other equivalence rela-
tions arising from dynamics, in particular homoclinicity and heteroclinicity. These
relations are in some sense transverse to orbit equivalence and the
C∗-algebras
which can be naturally associated to them are very different from the correspond-
ing crossed product. This can be observed already in what seems to be the earliest
of such constructions made by Wolfgang Krieger in [Kr2]: The C∗-algebras arising
from the homoclinic and heteroclinic equivalence relations of a mixing topological
Markov chain are simple AF-algebras while the corresponding crossed product is
neither AF nor simple.
David Ruelle was the next to construct a C∗-algebra from the equivalence rela-
tion given by homoclinicity in dynamical systems, cf. [Ru2], and it is his approach
I will consider in the present paper. What is crucial for the method of Ruelle is
that in many dynamical systems, such as the Smale spaces considered by Ruelle,
homoclinicity of two states can be extended to a ’uniform local homoclinicity’. See
Condition C of [Ru2]. It is this strengthening of the relation which ensures that the
topology on the graph of the equivalence relation defined by homoclinicity becomes
what is nowadays called an ´ etale equivalence relation, so that the construction of
Renault, [Re1], can be used to construct the
C∗-algebra
of the relation. Here I
take the stronger relation as point of departure and this allows the construction of
an ´ etale equivalence relation from the homoclinicity relation and the heteroclinicity
relation in more general settings than the Smale spaces introduced by Ruelle.
Besides the work of Krieger and Ruelle the paper builds on, and is strongly
influenced by the work of Ian Putnam and J. Wagoner. While Ruelle only con-
sidered homoclinicity, Putnam showed how one can construct the
C∗-algebras
of
heteroclinic equivalence in Smale spaces. For this he used the concept of a Haar
measure of the underlying groupoid, building again on the work of Renault. To let
go of the ´ etale condition is actually a weakening for many purposes, but through
his work with J. Spielberg, [PS], he was able to partly remedy this defect. One of
the major points of the present work is to show that an approach of Wagoner to
the construction of a dimension group representation for countable state Markov
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