# Spectral disjointness of rescalings of some surface flows

## TL;DR

This paper proves that for certain classes of surface flows, rescalings by different natural numbers are spectrally disjoint, indicating a form of spectral independence among these rescaled flows.

## Contribution

It establishes spectral disjointness for rescalings of special flows over circle rotations and interval exchange transformations with specific roof functions.

## Key findings

- Spectral disjointness holds for a full measure set of rotations.
- Spectral disjointness holds for a full measure set of IETs.
- Disjointness applies to flows with various types of roof functions.

## 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.