Self-Similar Inverse Semigroups from Wieler Solenoids

Abstract

Wieler showed that every irreducible Smale space with totally disconnected local stable sets is an inverse limit system, called a Wieler solenoid. We study self-similar inverse semigroups defined by s-resolving factor maps of Wieler solenoids. We show that the groupoids of germs and the tight groupoids of these inverse semigroups are equivalent to the unstable groupoids of Wieler solenoids. We also show that the C * -algebras of the groupoids of germs have a unique tracial state.We review the definition and some well-known properties about Smale spaces and their associated C * -algebras. We refer to [9-12] for more details.Definition 1 ([10]). Let (X, d) be a compact metric space and f : X → X a homeomorphism. Assume that we have constants 0 < λ < 1, 0 > 0 and a continuous map, called the bracket map,