On transfer operators on the circle with trigonometric weights

We study spectral properties of the transfer operators $L$ defined on the circle $\mathbb T=\mathbb R/\mathbb Z$ by $(Lu)(t)=\frac{1}{d}\sum_{i=0}^{d-1} f\left(\frac{t+i}{d}\right)u\left(\frac{t+i}{d}\right),\ t\in\mathbb T$ where $u$ is a function on $\mathbb T$. We focus in particular on the cases $f(t)=|\cos(πt)|^q$ and $f(t)=|\sin(πt)|^q$, which are closely related to some classical Fourier-analytic questions. We also obtain some explicit computations, particularly in the case $d=2$. Our study extends work of Strichartz \cite{Strichartz1990} and Fan and Lau \cite{FanLau1998}.

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