Markov partitions and homology for -solenoids

Burke, Nigel; Putnam, Ian F.

doi:10.1017/etds.2015.71

Cited by 3 publications

(4 citation statements)

References 15 publications

(31 reference statements)

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“…Theorem C is interesting for a few reasons. First, the computations given here generalize previous work by Burke and Putnam [BP17]. Second, the groupoids appearing in this context are not necessarily ample, providing us with examples where the HK conjecture holds beyond its original assumptions (the groupoid associated to an irrational flow on the torus provides another example, see Example 4.5 for details).…”

Section: Introductionsupporting

confidence: 57%

“…Degree 1 case. The case of K(c) = Q, that is, c ∈ Q, is considered by Burke and Putnam [BP17], where they computed the stable and unstable Putnam homology groups. To simplify the presentation, let us consider the case of two prime factors as follows.…”

mentioning

confidence: 99%

“…Remark 6.14. In view of Remark 3.3, the computation of Putnam's stable homology for (Y (c) , φ) with c ∈ Q carried out in [BP17] already gives equation (9). The method presented here is arguably more direct and does not require a deep understanding of the dynamics of the system; however, the method in [BP17] is intimately tied to Markov partitions and allows to understand the system in terms of symbolic coding.…”

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confidence: 99%

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Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

PROIETTI,

YAMASHITA

2024

Ergod. Th. Dynam. Sys.

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We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math.521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

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Section: Introductionsupporting

confidence: 57%

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confidence: 99%

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confidence: 99%

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Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

PROIETTI,

YAMASHITA

2024

Ergod. Th. Dynam. Sys.

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“…Nevertheless, in future work, the present authors will use the techniques from the present paper to compute the K-theory in a number of examples. In particular, the K-theory of the stable algebra of the p/q-solenoids studied in [3] will be computed.…”

Section: Introductionmentioning

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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Two New Roles in Norwich

Ward

2023

Springer Biographies

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Markov partitions and homology for -solenoids

Cited by 3 publications

References 15 publications

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

Homology and K-theory of dynamical systems IV. Further structural results on groupoid homology

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

Two New Roles in Norwich

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