Complete Determination of the Spectrum of a Transfer Operator associated with Intermittency

TL;DR

This paper provides a complete spectral analysis of the transfer operator associated with the Farey map, an example of an intermittent map, revealing its spectral decomposition and introducing an efficient numerical method for eigenvalue determination.

Contribution

It offers the first complete spectral analysis of the Farey map's transfer operator, including a proof of self-adjointness and a numerical method for eigenvalues.

Findings

Spectrum decomposes into continuous part and isolated eigenvalues

Transfer operator is self-adjoint on a specific Hilbert space

Numerical method effectively finds all eigenvalues, even in the continuous spectrum

Abstract

It is well established that the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point. We present here for the first time a complete spectral analysis for an example of such an intermittent map, the Farey map. We give a simple proof that the transfer operator is self-adjoint on a suitably defined Hilbert space and show that its spectrum decomposes into a continuous part (the interval $[0,1]$) and isolated eigenvalues of finite multiplicity. Using a suitable first-return map, we present a highly efficient numerical method for the determination of all the eigenvalues, including the ones embedded in the continuous spectrum.