The Metric Theory of Interval Exchange Transformations I. Generic Spectral Properties

Veech, William A.

doi:10.2307/2374396

Cited by 115 publications

(116 citation statements)

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“…'s are minimal (this is guaranteed by a condition due to Keane [Ke1], which is automatically dealt with if the interval lengths are rationally independent) but note that ergodic properties of minimal i.e.m. 's can differ substantially from those of circle rotations: first they need not be uniquely ergodic [Ke2,KN,Co], and second, being ergodic they can be weakly mixing [KS,V3,V4]. On the other hand uniquely ergodic i.e.m.…”

Section: Introductionmentioning

confidence: 98%

The cohomological equation for Roth-type interval exchange maps

Marmi

Moussa

Yoccoz³

2005

J. Amer. Math. Soc.

124

235

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We exhibit an explicit class of minimal interval exchange maps (i.e.m.’s) T T for which the cohomological equation \[ Ψ − Ψ ∘ T = Φ \Psi -\Psi \circ T=\Phi \] has a bounded solution Ψ \Psi provided that the datum Φ \Phi belongs to a finite codimension subspace of the space of functions having on each interval a derivative of bounded variation. The proof is purely dynamical and is based on a renormalization argument and on Gottshalk-Hedlund’s theorem. If the datum is more regular the loss of differentiability in solving the cohomological equation will be the same. The class of interval exchange maps is characterized in terms of a diophantine condition of Roth type imposed to an acceleration of the Rauzy-Veech-Zorich continued fraction expansion associated to T T . More precisely one must impose a growth rate condition for the matrices appearing in the continued fraction algorithm together with a spectral gap condition (which guarantees unique ergodicity) and a coherence condition. We also prove that the set of Roth-type interval exchange maps has full measure. In the appendices we construct concrete examples of Roth-type i.e.m.’s and we show how the growth rate condition alone does not imply unique ergodicity.

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Section: Introductionmentioning

confidence: 98%

The cohomological equation for Roth-type interval exchange maps

Marmi

Moussa

Yoccoz³

2005

J. Amer. Math. Soc.

124

235

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“…The same applies to the block of spacers immediately to its right: If this block of spacers is not the huge one, then its length is bounded by t 1 + · · · + t n0 , which is small compared to h n0 by (42). We finally get that the sum of the lengths of all non-huge blocks of spacers separating the building blocks in the decomposition of W is bounded by…”

Section: 2mentioning

confidence: 79%

“…This kind of spectral disjointness turns out to be sufficient to obtain a basic Möbius orthogonality lemma proved in [10]. As indicated in [9], the positive answer to Sarnak's conjecture in the class of all rank-one transformations would give automatically the positive answer to Sarnak's conjecture for almost every interval exchange transformations (see [42] for rank-one property of almost every IET). In [9] the 3-IET case is considered.…”

Section: 2mentioning

confidence: 82%

“…W n is subword of B n+1 . We then get by Lemma 13 that the length of the block of spacers immediately to the left of our copy of B n is at most t 1 + · · · + t n , which is small with respect to h n = |B n | by (42). In the same way, if B n appears in some W ′ n , ℓ ≤ n ≤ n 0 − 1, then the length of the block of spacers immediately to its right is bounded by the same quantity.…”

Section: 2mentioning

confidence: 96%

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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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“…First we shall use the duality between eigenvalues and the return times -going back at least to [15,27,20,30] -to show that, for any eigenvalue α of T t and any t := v|ω * with v ∈ Σ, we have…”

Section: Discrete Spectrummentioning

confidence: 99%