Eigenvalues and strong orbit equivalence

Cortéz, María Isabel; Durand, Fabien; Petite, Samuel

doi:10.1017/etds.2015.26

Cited by 8 publications

(18 citation statements)

References 29 publications

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“…Consequently, for any 0 ≤ θ < 1, such that e 2πiθ ∈ Sp(T ), there exists a clopen set whose measure is θ with respect to all invariant probability measures on X. Equivalently, Sp (T ) ⊆ τ ∈S(K 0 (X, T ), [1 X ]) τ (K 0 (X, T )), where S(K 0 (X, T ), [1 X ]) denotes the set of normalized traces (states) on K 0 (X, T ). This result has been recently stated in [4,Theorem 1. ] as well.…”

Section: Introductionsupporting

confidence: 64%

“…Let (X, T ) be a Cantor minimal system; as in [4], denote by E(X, T ) the subgroup of real numbers consisting of all θ such that exp(2πiθ) ∈ Sp (T ). This is the additive group of (continuous) eigenvalues.…”

Section: Spectra and Real Coboundariesmentioning

confidence: 99%

“…For the identification between J(X, T )/Θ(E(X, T )) and I(X, T )/E(X, T ) to be true, Inf (K 0 (X, T)) has to be trivial. In [4], an example of a Cantor minimal system for which neither Inf (K 0 (X, T)) nor the torsion subgroup of I(X, T )/E(X, T ) are trivial, is described.…”

Section: Corollary 211 ([4]mentioning

confidence: 99%

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Orbit equivalence of Cantor minimal systems and their continuous spectra

2017

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Abstract. To any continuous eigenvalue of a Cantor minimal system (X, T ), we associate an element of the dimension group K 0 (X, T ) associated to (X, T ). We introduce and study the concept of irrational miscibility of a dimension group. The main property of these dimension groups is the absence of irrational values in the additive group of continuous spectrum of their realizations by Cantor minimal systems. The strong orbit equivalence (respectively orbit equivalence) class of a Cantor minimal system associated to an irrationally miscible dimension group (G, u) (resp. with trivial infinitesimal subgroup) with trivial rational subgroup, have no non-trivial continuous eigenvalues.

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Section: Introductionsupporting

confidence: 64%

Section: Spectra and Real Coboundariesmentioning

confidence: 99%

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Orbit equivalence of Cantor minimal systems and their continuous spectra

2017

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“…This result complements the representation result of N. Ormes [Orm97, Theorem 6.1] that is used to prove that strong orbit equivalence of minimal Cantor systems is compatible with any group of eigenvalues as long as they have the same continuous eigenvalues that are roots of unity. For some deeper discussions and recent results on the relation of continuous eigenvalues with orbit equivalence see [CDP16,GHH16].…”

Section: 2mentioning

confidence: 99%

“…Proposition 25 in [CDP16] (see also [IO07]) asserts in specific cases that it is possible to realise some subgroups, but it is not clear which ones. Moreover, there are strong obstructions for this kind of realisations.…”

mentioning

confidence: 99%

Eigenvalues of minimal Cantor systems

2018

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In this article we give necessary and sufficient conditions that a complex number must satisfy to be a continuous eigenvalue of a minimal Cantor system. Similarly, for minimal Cantor systems of finite rank, we provide necessary and sufficient conditions for having a measure theoretical eigenvalue. These conditions are established from the combinatorial information of the Bratteli-Vershik representations of such systems. As an application, from any minimal Cantor system, we construct a strong orbit equivalent system without irrational eigenvalues which shares all measure theoretical eigenvalues with the original system. In a second application a minimal Cantor system is constructed satisfying the so-called maximal continuous eigenvalue group property.

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On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

et al. 2021

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Dimension groups are complete invariants of strong orbit equivalence for minimal Cantor systems. This paper studies a natural family of minimal Cantor systems having a finitely generated dimension group, namely the primitive unimodular proper S-adic subshifts. They are generated by iterating sequences of substitutions. Proper substitutions are such that the images of letters start with a same letter, and similarly end with a same letter. This family includes various classes of subshifts such as Brun subshifts or dendric subshifts, that in turn include Arnoux-Rauzy subshifts and natural coding of interval exchange transformations. We compute their dimension group and investigate the relation between the triviality of the infinitesimal subgroup and rational independence of letter measures. We also introduce the notion of balanced functions and provide a topological characterization of balancedness for primitive unimodular proper S-adic subshifts.

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Eigenvalues and strong orbit equivalence

Cited by 8 publications

References 29 publications

Orbit equivalence of Cantor minimal systems and their continuous spectra

Orbit equivalence of Cantor minimal systems and their continuous spectra

Eigenvalues of minimal Cantor systems

On the dimension group of unimodular $${\mathcal {S}}$$-adic subshifts

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