Systems of finite rank

Ferenczi, Sébastien

doi:10.4064/cm-73-1-35-65

Cited by 141 publications

(118 citation statements)

References 67 publications

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“…As observed by Ferenczi in [18], the symbolic system (Ω, T symb ) may in some cases be degenerate (for example with the von Neumann-Kakutani construction: B n+1 = B n B n ). However, it follows from the analysis developed by Kalikow in the annex of [28] that the factor map ψ is one-to-one, hence an isomorphism of the measure-preserving systems, whenever the limit sequence B ∞ := lim n→∞ B n is aperiodic.…”

Section: Remarkmentioning

confidence: 96%

“…The two classes of examples which we present here are known to be rigid, and weakly mixing (see [18]). 6.1.…”

Section: Examples Where the Cutting Parameter P N Is Unboundedmentioning

confidence: 99%

“…Assuming that the first tower is reduced to a single level P 1 = {F 1 }, as we can always do without loss of generality, the construction of the rank-one automorphism is then completely described by the sequences of parameters (p n ) n≥1 and (s n,i ) n≥1,0≤i≤pn−1 . There are many other ways to define rank-one transformations [18], [25], including symbolic models which are described in details in Section 7 (see also Section 4).…”

mentioning

confidence: 99%

“…[1], [2], [3], [4], [8], [14], [18], [30], [31], [34], [36], [37]), it still possesses many important open problems 1 . In this paper we will be mostly interested in the problem of spectral disjointness of positive 2 different powers T n , n ≥ 1, for a rank-one automorphism.…”

mentioning

confidence: 99%

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On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

Abdalaoui

Lemańczyk

Rue

2014

Journal of Functional Analysis

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Abstract. We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chacon's maps and Katok's map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of {U k T : k ∈ Z}, of "sufficiently many" analytic functions of the operator U T .Then we apply these disjointness results to prove Sarnak's conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rankone constructions: All sequences realized in these models are orthogonal to the Möbius function.Résumé. Nousétudions la disjonction spectrale des puissances d'une transformation de rang un. Pour une large classe de constructions de rang un, incluant celles dont les paramètres de découpage et empilage sont bornés, ainsi que d'autes exemples commes les transformations de Chacon généralisées et la transformation de Katok, nous prouvons que les puissances positives de la transformation sont deux-à-deux spectralement disjointes sur la partie continue du spectre. Notre preuve s'appuie sur l'existence, dans la fermeture faible de {U k T : k ∈ Z}, de suffisamment de fonctions analytiques de l'opérateur U T . Nous appliquons ensuite ces résultats de disjonction pour prouver la conjecture de Sarnak dans les modèles symboliques associésà ces constructions de rang un (qui peuvent ne pasêtre uniquement ergodiques) : toutes les suites réalisées dans ces modèles sont orthogonalesà la fonction de Möbius.

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Section: Remarkmentioning

confidence: 96%

“…The two classes of examples which we present here are known to be rigid, and weakly mixing (see [18]). 6.1.…”

Section: Examples Where the Cutting Parameter P N Is Unboundedmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

Abdalaoui

Lemańczyk

Rue

2014

Journal of Functional Analysis

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“…The flexibility and naturality of Vershik's construction is illustrated by the wide variety of dynamical systems they can be used to model, among others cutting and stacking constructions in ergodic theory (see Vershik's papers cited above, [Fer97]), substitution dynamical systems (see [Liv92], [LV92], [For97], [Hos00]), and interval exchange transformations (see [Fisa]). …”

Section: Introductionmentioning

confidence: 99%

Nonstationary Mixing and the Unique Ergodicity of Adic Transformations

Fisher¹

2009

Stoch. Dyn.

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Abstract. We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

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Entropy in Ergodic Theory

King

2009

Encyclopedia of Complexity and Systems Science

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Systems of finite rank

Cited by 141 publications

References 67 publications

On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

On spectral disjointness of powers for rank-one transformations and Möbius orthogonality

Nonstationary Mixing and the Unique Ergodicity of Adic Transformations

Entropy in Ergodic Theory

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