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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