Spectral analysis of transfer operators associated to Farey fractions

TL;DR

This paper investigates the spectral properties of transfer operators linked to the Farey map, revealing their self-adjointness, spectrum type, and special polynomial eigenfunctions for certain parameters.

Contribution

It provides a detailed spectral analysis of transfer operators for the Farey map, including self-adjointness and eigenfunction characterization for specific parameters.

Findings

Operators are self-adjoint on a suitable Hilbert space.

Spectrum is absolutely continuous with no non-zero point spectrum.

Polynomial eigenfunctions exist for negative half-integer parameters.

Abstract

The spectrum of a one-parameter family of signed transfer operators associated to the Farey map is studied in detail. We show that when acting on a suitable Hilbert space of analytic functions they are self-adjoint and exhibit absolutely continuous spectrum and no non-zero point spectrum. Polynomial eigenfunctions when the parameter is a negative half-integer are also discussed.