  We prove structural results for measure preserving systems, called
Furstenberg systems, naturally associated with bounded multiplicative
functions. We show that for all pretentious multiplicative functions these
systems always have rational discrete spectrum and, as a consequence, zero
entropy. We obtain several other refined structural and spectral results, one
consequence of which is that the Archimedean characters are the only
pretentious multiplicative functions that have Furstenberg systems with trivial
rational spectrum, another is that a pretentious multiplicative function has
ergodic Furstenberg systems if and only if it pretends to be a Dirichlet
character, and a last one is that for any fixed pretentious multiplicative
function all its Furstenberg systems are isomorphic. We also study structural
properties of Furstenberg systems of a class of multiplicative functions,
introduced by Matom\"aki, Radziwill, and Tao, which lie in the intermediate
zone between pretentiousness and strong aperiodicity. In a work of the last two
authors and Gomilko, several examples of this class with exotic ergodic
behavior were identified, and here we complement this study and discover some
new unexpected phenomena. Lastly, we prove that Furstenberg systems of general
bounded multiplicative functions have divisible spectrum. When these systems
are obtained using logarithmic averages, we show that trivial rational spectrum
implies a strong dilation invariance property, called strong stationarity, but,
quite surprisingly, this property fails when the systems are obtained using
Ces\`aro averages.
