Almost periodic currents, chains and divisors in tube domains

Abstract

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.