Nonconventional ergodic theorems for quantum dynamical systems

Abstract

We study the so-called Nonconventional Ergodic Theorem for noncommutative generic measures introduced by Furstenberg in classical ergodic theory, and the relative three-point multiple correlations of arbitrary length arising from several situations of interest in quantum case. We deal with the diagonal state canonically associated to the product state (i.e. quantum "diagonal measures") in the ergodic situation, and with the case concerning convex combinations (i.e. direct integral) of diagonal measures in nonergodic one. We also treat in the full generality the case of compact dynamical systems, that is when the unitary generating the dynamics in the Gelfand–Naimark–Segal representation is almost periodic. In all the above-mentioned situations, we provide the explicit formula for the involved ergodic average. Such an explicit knowledge of the limit of the three-point correlations is naturally relevant for the investigation of the long time behavior of a dynamical system.