  We study irreducible Smale spaces with totally disconnected stable sets and
their associated $K$-theoretic invariants. Such Smale spaces arise as Wieler
solenoids, and we restrict to those arising from open surjections. The paper
follows three converging tracks: one dynamical, one operator algebraic and one
$K$-theoretic. Using Wieler's Theorem, we characterize the unstable set of a
finite set of periodic points as a locally trivial fibre bundle with discrete
fibres over a compact space. This characterization gives us the tools to
analyze an explicit groupoid Morita equivalence between the groupoids of
Deaconu-Renault and Putnam-Spielberg, extending results of Thomsen. The
Deaconu-Renault groupoid and the explicit Morita equivalence leads to a
Cuntz-Pimsner model for the stable Ruelle algebra. The $K$-theoretic invariants
of Cuntz-Pimsner algebras are then studied using the Pimsner extension, for
which we construct an unbounded representative. To elucidate the power of these
constructions we characterize the KMS weights on the stable Ruelle algebra of a
Wieler solenoid. We conclude with several examples of Wieler solenoids, their
associated algebras and spectral triples.
