On 3-dimensional foliated dynamical systems and Hilbert type reciprocity law.

We introduce a geometric analog of the Hilbert symbol and show a Hilbert type reciprocity law for a 3-dimensional foliated dynamical system (FDS3 for short). This answers the question posed by Deninger. For this, we employ the theory of smooth Deligne cohomology and the integration theory of Deligne cohomology classes. We also present a structure theorem for an FDS3, which yields a classification of FDS3's, and we construct concrete examples of FDS3's for each type of the classification. [ABSTRACT FROM AUTHOR]