On spectral disjointness of powers for rank-one transformations and M\"obius orthogonality

TL;DR

This paper proves spectral disjointness of different powers of certain rank-one transformations and applies these results to confirm Sarnak's conjecture, showing orthogonality to the Möbius function in related symbolic models.

Contribution

It establishes spectral disjointness for a broad class of rank-one transformations and verifies Sarnak's conjecture for their symbolic models.

Findings

Different positive powers are spectrally disjoint on the continuous spectrum.

Spectral disjointness leads to orthogonality to the Möbius function.

Results apply to various examples including Chacon's maps and Katok's map.

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_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 M\"obius function.