CANTORI SPECTRA FOR THE SAWTOOTH MAP

We calculate analytically the Fourier spectrum for the cantori of the sawtooth maps. These maps are a one-parameter family of chaotic area-preserving maps. We show that the Fourier spectrum grows exponentially for parameters close to criticality, and that it exhibits self-similarity structure at all length scales. The self-similarity scales as the quotients of successive denominators of the convergents of irrational numbers. We compute exactly the scaling for quadratic irrationals. The behaviour of the spectrum for large values of the perturbation parameter is also investigated.