Spectral disjointness of rescalings of some surface flows

Abstract: We study self-similarity problem for two classes of flows: (1) special flows over circle rotations and under roof functions with symmetric logarithmic singularities (2) special flows over interval exchange transformations and under roof functions which are of two types * piecewise constant with one additional discontinuity which is not a discontinuity of the IET; * piecewise linear over exchanged intervals with non-zero slope. We show that if ${T^f_t}_{t\in\mathbb R}$ is as in (1) then for a full measure set of rotations, and for every two distinct natural numbers $K$ and $L$, we have that ${T^f_{Kt}}_{t\in\mathbb R}$ and ${T^f_{Lt}}_{t\in\mathbb R}$ are spectrally disjoint. Similarly, if ${T^f_t}_{t\in\mathbb R}$ is as in (2), then for a full measure set of IET's, a.e. position of the additional discontinuity (of $f$, in piecewise constant case) and every two distinct natural numbers $K$ and $L$, the flows ${T^f_{Kt}}_{t\in\mathbb R}$ and ${T^f_{Lt}}_{t\in\mathbb R}$ are spectrally disjoint.