Quantized chaotic dynamics and non-commutative KS entropy

We study the quantization of two examples of classically chaotic dynamics, the Anosov dynamics of “cat maps” on a two dimensional torus, and the dynamics of baker’s maps. Each of these dynamics is implemented as a discrete group of automorphisms of a von Neumann algebra of functions on a quantized torus. We compute the non- commutative generalization of the Kolmogorov-Sinai entropy, namely the Connes-Stormer entropy, of the generator of this group, and find that its value is equal to the classical value. This can be interpreted as a sign of persistence of chaotic behavior in a dynamical system under quantization.