  We develop a semi-classical approximation for the scar function in the
  Weyl-Wigner representation in the neighborhood of a classically unstable
periodic orbit of chaotic two dimensional systems. The prediction of hyperbolic
fringes, asymptotic to the stable and unstable manifolds, is verified
computationally for a (linear) cat map, after the theory is adapted to a
discrete phase space appropriate to a quantized torus. Characteristic fringe
patterns can be distinguished even for quasi-energies where the fixed point is
not Bohr-quantized. Also the patterns are highly localized in the neighborhood
of the periodic orbit and along its stable and unstable manifolds without any
long distance patterns that appears for the case of the spectral Wigner
function.
