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## Abstract

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 quantized linked twist maps on the torus. In the appendix, 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.

Translated title of the contribution | Weyl's law and quantum ergodicity for maps with divided phase space |
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Original language | English |

Pages (from-to) | 277 - 304 |

Number of pages | 18 |

Journal | Nonlinearity |

Volume | 18 (1) |

DOIs | |

Publication status | Published - Jan 2005 |

### Bibliographical note

Publisher: IOP Publishing LtdOther identifier: IDS Number: 894JQ

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