  In this paper we construct a sequence of eigenfunctions of the ``quantum
Arnold's cat map'' that, in the semiclassical limit, show a strong scarring
phenomenon on the periodic orbits of the dynamics. More precisely, those states
have a semiclassical limit measure that is the sum of 1/2 the normalized
Lebesgue measure on the torus plus 1/2 the normalized Dirac measure
concentrated on any a priori given periodic orbit of the dynamics. It is known
(the Schnirelman theorem) that ``most'' sequences of eigenfunctions
equidistribute on the torus. The sequences we construct therefore provide an
example of an exception to this general rule. Our method of construction and
proof exploits the existence of special values of Planck's constant for which
the quantum period of the map is relatively ``short'', and a sharp control on
the evolution of coherent states up to this time scale. We also provide a
pointwise description of these states in phase space, which uncovers their
``hyperbolic'' structure in the vicinity of the fixed points and yields more
precise localization estimates.
