  We prove a structural result for measure preserving systems naturally
associated with any finite collection of multiplicative functions that take
values on the complex unit disc. We show that these systems have no irrational
spectrum and their building blocks are Bernoulli systems and infinite-step
nilsystems. One consequence of our structural result is that strongly aperiodic
multiplicative functions satisfy the logarithmically averaged variant of the
disjointness conjecture of Sarnak for a wide class of zero entropy topological
dynamical systems, which includes all uniquely ergodic ones. We deduce that
aperiodic multiplicative functions with values plus or minus one have
super-linear block growth. Another consequence of our structural result is that
products of shifts of arbitrary multiplicative functions with values on the
unit disc do not correlate with any totally ergodic deterministic sequence of
zero mean. Our methodology is based primarily on techniques developed in a
previous article of the authors where analogous results were proved for the
M\"obius and the Liouville function. A new ingredient needed is a result
obtained recently by Tao and Ter\"av\"ainen related to the odd order cases of
the Chowla conjecture.
