On transfer operators on the circle with trigonometric weights

TL;DR

This paper investigates the spectral properties of transfer operators on the circle with trigonometric weights, providing explicit computations and extending previous work related to Fourier analysis and classical questions.

Contribution

It offers new insights into the spectral analysis of transfer operators with specific trigonometric weights, including explicit cases for $d=2$, extending prior foundational studies.

Findings

Spectral properties characterized for weights $| ext{cos}( ext{pi} t)|^q$ and $| ext{sin}( ext{pi} t)|^q$

Explicit computations achieved for the case $d=2$

Extended classical Fourier-analytic questions through spectral analysis

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}.