Homology and $K$-theory of dynamical systems II. Smale spaces with totally disconnected transversal

Abstract

We apply our previous work on the relation between groupoid homology and $K$-theory to Smale spaces. More precisely, we consider the unstable equivalence relation of a Smale space with totally disconnected stable sets and prove that the associated spectral sequence shows Putnam's stable homology groups on the second sheet. Moreover, this homology is in fact isomorphic to the groupoid homology of the unstable equivalence relation.