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

El Houcein El Abdalaoui (1) , Mariusz Lemanczyk (2) , Thierry de La Rue (1)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Faculty of Mathematics and Computer Science

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_T^k: k in 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 rank-one constructions: All sequences realized in these models are orthogonal to the Möbius function.

Mots clés: rank-one automorphisms; spectral disjointness; Möbius function; Sarnak's conjecture

Domaines: Systèmes dynamiques [math.DS]; Théorie des nombres [math.NT]

El Houcein El Abdalaoui, Mariusz Lemanczyk, Thierry de La Rue. On spectral disjointness of powers for rank-one transformations and Möbius orthogonality. Journal of Functional Analysis, 2014, 266 (1), pp.284-317. ⟨10.1016/j.jfa.2013.09.005⟩. ⟨hal-00769294⟩