Egorov property in perturbed cat map

We study the time evolution of the quantum-classical correspondence (QCC) for the well known model of quantised 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 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 in particular it decays faster then exponential. Here we study and explain the observed behavior of the QCF and the apparent violation of the QCC principle.