On transfer operators on the circle with trigonometric weights

Abstract

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}.

30 pages, 4 figures