MEAN DYNAMICAL ENTROPY OF QUANTUM MAPS ON THE SPHERE DIVERGES IN THE SEMICLASSICAL LIMIT

We analyze quantum dynamical entropy based on the notion of coherent s tales. The mean value of this quantity for quantum maps on the sphere is computed as an average over the uniform measure on the space of uni tary matrices of size N. Mean dynamical entropy is positive for N grea ter than or equal to 3, which supplies a direct link between random ma trices of the circular unitary ensemble and the chaotic dynamics of th e corresponding classical maps. Mean entropy tends logarithmically to infinity in the semiclassical limit N --> infinity and this indicates the ubiquity of chaos in classical mechanics.