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 $f(t)=|\cos(\pi t)|^q$ and $f(t)=|\sin(\pi t)|^q$, 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}.
Categories: math.DS math.CA math.NT