  A notion of almost periodic current is introduced, as well as a notion of
almost periodic holomorphic chain proceeded from that definition. Such a chain
can be defined either as a special case of almost periodic currents or as a
holomorphic chain whose trace measure is an almost periodic distribution. It is
shown that in general situation almost periodicity of the trace of a current
does not imply that for the current itself, even if it is closed and positive.
  The zero set (regarded as a holomorphic chain) of a holomorphic mapping can
be represented as a Monge-Ampere type current, and one could expect that the
zero set of an almost periodic holomorphic mapping should be almost periodic;
however we construct an example of an almost periodic holomorphic mapping whose
zero set is not almost periodic. Nevertheless, we prove almost periodicity of
the Monge-Ampere currents corresponding to almost periodic holomorphic mappings
with certain additional properties. Then we construct functions that play the
same role for almost periodic divisors as the so-called Jessen functions for
almost periodic holomorphic functions. In terms of Jessen function we give a
sufficient condition for realizability of an almost periodic divisor as the
divisor of a holomorphic almost periodic function; some necessary condition is
obtained, too.
