  For a general class of unitary quantum maps, whose underlying classical phase
space is divided into several invariant domains of positive measure, we
establish analogues of Weyl's law for the distribution of eigenphases. If the
map has one ergodic component, and is periodic on the remaining domains, we
prove the Schnirelman-Zelditch-Colin de Verdiere Theorem on the
equidistribution of eigenfunctions with respect to the ergodic component of the
classical map (quantum ergodicity). We apply our main theorems to quantised
linked twist maps on the torus. In the Appendix, S. Zelditch connects these
studies to some earlier results on `pimpled spheres' in the setting of
Riemannian manifolds. The common feature is a divided phase space with a
periodic component.
