  In this Article, several aspects of the asymptotic dynamics of
finite-dimensional open quantum systems are explored. First, after recalling a
structure theorem for the peripheral map, we discuss sufficient conditions and
a characterization for its unitarity. Interestingly, this is not always
guaranteed due to the presence of permutations in the structure of the
asymptotic map. Then, we show the connection between the asymptotic map and the
modular theory by Tomita and Takesaki.
