Spectral orthogonality of special flows

TL;DR

This paper investigates spectral orthogonality in special flows over irrational rotations with analytic or piecewise $C^1$ roof functions, showing conditions under which flows are spectrally orthogonal based on their properties.

Contribution

It establishes spectral orthogonality results for von-Neumann flows with different roof functions, extending understanding of spectral properties in these dynamical systems.

Findings

Spectral orthogonality holds for weak-mixing flows with analytic roof functions for a dense set of parameters.

For piecewise $C^1$ roof functions, spectral orthogonality occurs for a full measure set of parameters.

Flows with different parameters are often spectrally orthogonal under specified conditions.

Abstract

In this paper, we study the spectral orthogonality problem for special flows built over irrational rotations under two different types of roof functions: 1) the roof functions are real analytic. 2) the roof functions are piecewise $C^1$ with one discontinuity. These flows are also known as von-Neumann flows. We show that if $\{T^f_\alpha\}$ is as in 1) and weak mixing, then for a $G_\delta$ dense set of $\beta$, we have that $\{T^f_\beta\}$ is weak-mixing and is spectrally orthogonal to $\{T^f_\alpha\}$. On the other hand, if $\{T^f_\alpha\}$ is as in 2), then for a full measure set of $\beta$, the flows $\{T^f_\alpha\}$ and $\{T^f_\beta\}$ are spectrally orthogonal.