  We develop a semiclassical approximation for the spectral Wigner and Husimi
functions in the neighbourhood of a classically unstable periodic orbit of
chaotic two dimensional maps. The prediction of hyperbolic fringes for the
Wigner function, 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. The characteristic
fringe patterns can be distinguished even for quasi-energies where the fixed
point is not Bohr-quantized. The corresponding Husimi function dampens these
fringes with a Gaussian envelope centered on the periodic point. Even though
the hyperbolic structure is then barely perceptible, more periodic points stand
out due to the weakened interference.
