The Homoclinic and Heteroclinic C-algebras of a Generalized One-dimensional Solenoid

The homoclinic and heteroclinic structure in dynamical systems was first used to produce C-algebras in the way breaking work of Cuntz and Krieger in [Kr1], [Kr2] and [CuK]. This work has been generalized in many directions where the relation to dynamical systems is either absent or appears very implicit, but Ian Putnam described in [Pu] a natural way to extend the constructions of Cuntz and Krieger to higher dimensions such that the point of departure is the heteroclinic structure in a Smale space, just as the work of Cuntz and Krieger departed from the heteroclinic structure in a shift of finite type, which is a zero-dimensional Smale space. Putnam builds his approach on the work of D. Ruelle, [Ru1], [Ru2], who introduced the notion of a Smale space in [Ru1] and constructed the so-called asymptotic algebra from the homoclinic equivalence relation in [Ru2]. The work of Putnam and Ruelle was further generalized by the author in [Th1] where it was shown that Ruelle’s approach can be adopted as soon as there is enough expansiveness in the underlying dynamical system; the local product structure in a Smale space is not crucial for the construction. Furthermore, in [Th1] the alternative approach was used to obtain inductive limit decompositions for the algebras of Putnam arising from particular classes of Smale spaces, e.g. expansive group automorphisms and one-dimensional generalized solenoids in the sense of R.F. Williams, [Wi2], and I. Yi, [Y1]. For expansive group automorphisms it was shown that the C-algebras are all AT-algebras of real rank zero, and hence are classified by their K-theory groups, thanks to the work of G. Elliott, [Ell1]. For one-dimensional generalized solenoids the exact nature of the inductive limit decomposition was not determined and the homoclinic algebra was not examined. In particular, it was not decided if the C-algebras are classified by K-theory. The main purpose of the present paper is tie up this loose end by showing that they are, although they turn out to be more general AH-algebras and exhibit more complicated K-theory than the algebras arising from expansive group automorphisms, at least in the sense that torsion appears. Specifically, it is shown that the heteroclinic algebra of both a one-solenoid and its inverse, as well as the homoclinic algebra are all AH-algebras of real rank zero with no dimension growth. They are therefore classified by K-theory thanks to the work of Elliott and Gong, [EG]. This conclusion is obtained for the heteroclinic algebra by combining a thorough study of the inductive limit decomposition obtained in [Th1] with results on the classification of simple C-algebras, in particular results by H. Lin on algebras of tracial rank zero, cf. e.g. [Lin4].