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

Abstract

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.

0-1 sequences of the Thue-Morse type and Sarnak’s conjecture

“…However, in some cases this idea cannot be applied directly (for example, if the systems under consideration are not weakly mixing, which is the case of the present article), the spectral approach does help, see recent [1], [3]. This approach -a mixture of spectral disjointness with a direct proof of the Möbius orthogonality of the sequences corresponding to the discrete part of the spectrum -will also be applied here.…”

“…From now on, only strictly ergodic case is considered. 1 We will consider Sarnak's conjecture [23] which says that whenever S is a zero (topological) entropy homeomorphism of a compact metric space Y then, for each f ∈ C(Y ) and y ∈ Y , we have (2) 1…”

“…In numerous cases, this conjecture has recently been proved to hold: [1], [3], [4], [9], [16], [19], [21] (see also [25]). …”

“…From this point of view, the present article can be seen as first steps to handle both these problems. Borrowing an idea from [1], we consider Sarnak's conjecture (2) …”

“…Basic spectral theory for group extensions shows that the spectral measure σ fm of f m is absolutely continuous with respect to σ, so by Remark 2 and an observation from [1],…”