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

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 Chaconʼs maps and Katokʼs 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 {UTk:k∈Z}, of “sufficiently many” analytic functions of the operator UT. Then we apply these disjointness results to prove Sarnakʼs 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.