  We study semi-classical limits of eigenfunctions of a quantized linear
hyperbolic automorphism of the torus ("cat map"). For some values of Planck's
constant, the spectrum of the quantized map has large degeneracies. Our first
goal in this paper is to show that these degeneracies are coupled to the
existence of quantum symmetries. There is a commutative group of unitary
operators on the state-space which commute with the quantized map and therefore
act on its eigenspaces. We call these "Hecke operators", in analogy with the
setting of the modular surface.
  We call the eigenstates of both the quantized map and of all the Hecke
operators "Hecke eigenfunctions". Our second goal is to study the semiclassical
limit of the Hecke eigenfunctions. We will show that they become
equidistributed with respect to Liouville measure, that is the expectation
values of quantum observables in these eigenstates converge to the classical
phase-space average of the observable.
