Ergodicity and scars of the quantum cat map in the semiclassical regime - art. no. 056208

We study the quantum localization effect of the cat map manifested in the m otions of statistical ensembles. Specifically, the coarse-grained entropy a nd time-averaged phase space distributions are investigated. For this purpo se, an amended version of the Wigner function on the discretized phase toru s is presented. We find that the time average of the coarse-grained Wigner function is scarred (antiscarred) along some short periodic orbits, and the heights (depths) of these scars (antiscars) decrease in a linear way with the Planck constant when the semiclassical limit is approached. The relatio nship between the scars observed here and those exhibited in the quasienerg y eigenstates is discussed.