Influence of scars on time averaged ensemble distribution of quantum cat map in the semiclassical regime

We study nonergodic behavior of an ensemble of quantum cat map with initially localized phase-space distributions. For this purpose, an amended version of the Wigner function on discretized phase torus is given, which shows meaningful classical-quantum correspondence in the semiclassical limit. We find the time averaged coarse-grained Wigner function is scarred and antiscared along some short periodic orbits, and the heights (depths) of these scars (antiscars) decrease linearly with (h) over bar when the initial phase-space distribution is fixed and the semiclassical limit is taken by letting (h) over bar = 1/(2 pi N(0)b(l)) with b a prime and l --> infinity.