Phase-space localization: Topological aspects of quantum chaos

We study quantized classically chaotic maps on a toroidal two-diensional phase space. A discrete, topological criterion for phase-space localization is presented. To each eigenfunction an integer is associated, analogous to a quantized Hall conductivity, which when nonzero reflects phase-space delocalization. A model system is studied, and a correspondence between delocalization and chaotic classical dynamics is discussed.