Spectral orthogonality of special flows
Mingcheng Sheng
TL;DR
The work addresses spectral disjointness for special flows over irrational rotations with two roof regimes: analytic roof functions and von Neumann-type roofs with a single discontinuity. It employs a rigidity–mixing criterion to prove spectral orthogonality: for Liouvillean $eta$ and analytic roof $f$ (when $T_eta^f$ is weakly mixing), a $G_ abla$-dense set of $oldsymbol{α}$ yields $T_oldsymbol{α}^f$ that is weakly mixing and spectrally orthogonal to $T_eta^f$, while for von Neumann flows a full-measure set of $oldsymbol{β}$ (relative to fixed $oldsymbol{α}$) gives spectral orthogonality. The analysis combines Fourier methods, uniform stretch arguments, and continued-fraction techniques to construct rigidity sequences along convergents and to establish mixing along corresponding subsequences. In the von Neumann case, the authors derive a full-measure orthogonality result by exploiting Ostrowski expansions and a measure-theoretic argument showing that the exceptional set has measure zero. Overall, the paper advances understanding of spectral disjointness for reparameterizations of linear toral flows and provides a versatile framework for proving orthogonality in both analytic and discontinuous-roof settings.
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 $\{T^f_α\}$ is as in 1) and weak mixing, then for a $G_δ$ dense set of $β$, we have that $\{T^f_β\}$ is weak-mixing and is spectrally orthogonal to $\{T^f_α\}$. On the other hand, if $\{T^f_α\}$ is as in 2), then for a full measure set of $β$, the flows $\{T^f_α\}$ and $\{T^f_β\}$ are spectrally orthogonal.