The Egorov property in perturbed cat maps

We study the time evolution of the quantum-classical correspondence (QCC) for the well-known model of quantized perturbed cat maps on the torus in the very specific regime of semi-classically small perturbations. The quality of the QCC is measured by the overlap of the classical phase-space density and corresponding Wigner function of the quantum system called quantum-classical fidelity (QCF). In the analysed regime the QCF strongly deviates from the known general behaviour discussed in Horvat et al (2006 Nonlinearity 19 1-23 (Preprint quant-ph/0601139)), in particular it decays faster than exponential. Here we study and explain the observed behaviour of the QCF and the apparent violation of the QCC principle