  We show some fundamental results concerning $3$-dimensional foliated
dynamical systems (FDS$^3$ for short) introduced by Deninger. Firstly, we give
a decomposition theorem for an FDS$^3$, which yields a classification of
FDS$^3$'s. Secondly, for each type of the classification, we construct concrete
examples of FDS$^3$'s. Finally, by using the integration theory for smooth
Deligne cohomology, we introduce geometric analogues of local symbols and show
a Hilbert type reciprocity law for an FDS$^3$. Our results answer the question
posed by Deninger.
