Spectral behavior of contractive noise

We study the behavior of the spectra corresponding to quantum systems subjected to a contractive noise, i.e. the environment reduces the accessible phase space of the system, but the total probability is conserved. We find that the number of long lived resonances grows as a power law in $\hbar$ but surprisingly there is no relationship between the exponent of this power law and the fractal dimension of the corresponding classical attractor. This is in disagreement with the predictions of the fractal Weyl law which has been established for open systems where the probability is lost under the effect of a projective noise.