The homoclinic and heteroclinic <i>C</i><sup>*</sup>-algebras of a generalized one-dimensional solenoid

Thomsen, Klaus

doi:10.1017/s0143385709000042

Cited by 13 publications

(22 citation statements)

References 33 publications

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“…(We remark that this description of ρ is almost identical to that given in [Th5,Lemma 3.5] in a different setting.) Define p j : D j → A j such that p j (a, f ) = a, and note that p k+1 • is homotopic to (χ + µ • I k ) • p k so that…”

Section: Markov Mapssupporting

confidence: 70%

Circle maps and -algebras

Schmidt¹,

Thomsen²

2013

Ergod. Th. Dynam. Sys.

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We consider a construction of C * -algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the C * -algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K -theory groups and turn them into an algorithmic method for Markov maps.

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Section: Markov Mapssupporting

confidence: 70%

Circle maps and -algebras

Schmidt¹,

Thomsen²

2013

Ergod. Th. Dynam. Sys.

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“…We illustrate the techniques for doing this in the example of the 'aab/ab-solenoid', a specific example of a Williams solenoid constructed from a certain map on a wedge sum of two circles. The reader should be aware that the K -theory of this example (and any one-dimensional Williams solenoid) is well known, see [33] (and also [24,25]). Our techniques are also applicable to any one-dimensional Williams solenoid.…”

Section: Theorem 12 (See Theorem 417)mentioning

confidence: 99%

“…Note that g is not a local homeomorphism in this example. For more details on this specific example and one-solenoids in general, see [25,28,33].…”

Section: Examples Of Wieler Solenoidsmentioning

confidence: 99%

The stable algebra of a Wieler solenoid: inductive limits and -theory

Deeley¹,

Yashinski²

2019

Ergod. Th. Dynam. Sys.

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Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable C * -algebra is the stationary inductive limit of a C * -stable Fell algebra that has compact spectrum and trivial Dixmier-Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to in principle compute the K-theory of the stable C * -algebra. A specific onedimensional Smale space (the aab/ab-solenoid) is considered as an illustrative running example throughout. 1 2 ROBIN J. DEELEY AND ALLAN YASHINSKIunstably equivalent to some point in P. On the set X u (P), we consider the stable equivalence relation ∼ s , viewed as a groupoidThe groupoid G s (P) has anétale topology, and the stable C * -algebra of (X, ϕ) is the groupoid C * -algebra C * (G s (P)).In the case where X is a Wieler solenoid, we use the inverse limit structure to define a subrelation ∼ 0 of ∼ s . There is a corresponding subgroupoid (P), and therefore G 0 (P) isétale. Building on this, we use the fact that the inverse limit is stationary to prove the following result.Theorem 0.1. There is a nested sequence ofétale subgroupoidsThis allows one to reduce the study of G s (P) to G 0 (P), which is easier to understand. To see why it is easier, first note that the space X u (P) has a natural topology, which coincides with the topology of the diagonal subspace of G s (P). Note we never consider the subspace topology of X u (P) it inherits from X, as X u (P) is dense as a subset of X. Likewise the topology defined on G s (P) is not the same as the subspace topology it inherits as a subspace of X u (P) × X u (P). However, the topology of G 0 (P) does coincide with the subspace topology from

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“…It was shown by Williams that expanding attractors of certain diffeomorphisms of compact manifolds are one-solenoids via a conjugacy which turns the restriction of the diffeomorphism into f . He also showed that each one-solenoid arises in this way from a diffeomorphism of the 4-sphere [11].…”

Section: One-dimensional Generalized Solenoidsmentioning

confidence: 99%

“…Following Williams and Yi [15,17], Thomsen called (X, f ) a generalized one-dimensional solenoid or just a one-solenoid [11]. It was shown by Williams that expanding attractors of certain diffeomorphisms of compact manifolds are one-solenoids via a conjugacy which turns the restriction of the diffeomorphism into f .…”

Section: One-dimensional Generalized Solenoidsmentioning

confidence: 99%