On the leading eigenvalue of transfer operators of the Farey map with real temperature

TL;DR

This paper investigates the spectral properties of transfer operators related to the Farey map, demonstrating their self-adjointness and providing a method to approximate the leading eigenvalue using matrix representations.

Contribution

It introduces a framework for analyzing the spectral properties of transfer operators of the Farey map on holomorphic function spaces, including self-adjointness and eigenvalue approximation techniques.

Findings

Operators are self-adjoint on a suitable space of holomorphic functions

The positive dominant eigenvalue can be approximated via matrix representations

Provides insights into the spectral structure of Farey map transfer operators

Abstract

We study the spectral properties of a family of generalized transfer operators associated to the Farey map. We show that when acting on a suitable space of holomorphic functions, the operators are self-adjoint and the positive dominant eigenvalue can be approximated by means of the matrix expression of the operators.