On the generalised transfer operators of the Farey map with complex temperature

TL;DR

This paper investigates the spectral properties of generalized transfer operators of the Farey map, linking them to the Selberg Zeta function and dynamical zeta functions, with a focus on eigenvalues and operator analysis.

Contribution

It formulates the eigenvalue problem for these operators on a Hilbert space and reduces it to a linear algebra problem involving infinite matrices, advancing spectral theory understanding.

Findings

Established the eigenvalue problem for generalized transfer operators

Connected spectral properties to the Selberg Zeta function

Reduced the problem to linear algebra on infinite matrices

Abstract

We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This problem is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices.