Weyl’s Law and Quantum Ergodicity for Maps with Divided Phase Space

For a general class of unitary quantum maps, whose underlying classical phase space is divided into ergodic and non-ergodic components, we prove analogues of Weyl’s law for the distribution of eigenphases, and the Schnirelman-Zelditch-Colin de Verdière Theorem on the equidistribution of eigenfunctions with respect to the ergodic components of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus.