Classical and quantum ergodicity on orbifolds

TL;DR

This paper extends quantum ergodicity results to orbifolds, showing that ergodic Hamiltonian flows imply quantum ergodicity for certain operators, and proves ergodicity of geodesic flow on negatively curved orbifolds.

Contribution

It generalizes classical quantum ergodicity theorems to the setting of orbifolds and establishes ergodicity of geodesic flow on negatively curved orbifolds.

Findings

Quantum ergodicity holds on orbifolds under ergodic Hamiltonian flow.

Ergodicity of geodesic flow is proven for negatively curved orbifolds.

Extension of classical results to orbifold setting.

Abstract

We extend to orbifolds classical results on quantum ergodicity due to Shnirelman, Colin de Verdi\`ere and Zelditch, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, ergodicity of the Hamiltonian flow of p implies quantum ergodicity for the operator P. We also prove ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.