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

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

2001-08-24

arxiv.org/abs/nlin/0108044v1

Nonlinear Sciences

Chaotic Dynamics

4 pages, 3 figures included

Scientific paper

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.

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