Quantized Chaotic Dynamics and Non-commutative KS Entropy

Abstract 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.