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

Weyl's law and quantum ergodicity for maps with divided phase space (with an appendix Converse quantum ergodicity)