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.
Highlights
The purpose of this work is to study groupoids of germs and tight groupoids on a certain class ofSmale spaces
Wieler [1] showed that irreducible Smale spaces with totally disconnected local stable sets can be realized as stationary inverse limit systems satisfying certain conditions, called Wieler solenoids [1,2,3,4]
If ( X, f ) is an irreducible Wieler solenoid, there is an irreducible subshift of finite type (Σ, σ ) and an s-resolving factor map π : (Σ, σ ) → ( X, f )
Summary
Wieler [1] showed that irreducible Smale spaces with totally disconnected local stable sets can be realized as stationary inverse limit systems satisfying certain conditions, called Wieler solenoids [1,2,3,4]. Nekrashevych defined self-similar inverse semigroups, called adjacency semigroups, on Smale spaces with s-resolving factor maps [5,6,7]. The limit solenoid defined by the adjacency semigroup of a Smale space is topologically conjugate to the original. We use adjacency semigroups to study unstable equivalence relations on Wieler solenoids.
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