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We prove structural results for measure preserving systems, calledFurstenberg systems, naturally associated with bounded multiplicativefunctions. We show that for all pretentious multiplicative functions thesesystems always have rational discrete spectrum and, as a consequence, zeroentropy. We obtain several other refined structural and spectral results, oneconsequence of which is that the Archimedean characters are the onlypretentious multiplicative functions that have Furstenberg systems with trivialrational spectrum, another is that a pretentious multiplicative function hasergodic Furstenberg systems if and only if it pretends to be a Dirichletcharacter, and a last one is that for any fixed pretentious multiplicativefunction all its Furstenberg systems are isomorphic. We also study structuralproperties of Furstenberg systems of a class of multiplicative functions,introduced by Matom\"aki, Radziwill, and Tao, which lie in the intermediatezone between pretentiousness and strong aperiodicity. In a work of the last twoauthors and Gomilko, several examples of this class with exotic ergodicbehavior were identified, and here we complement this study and discover somenew unexpected phenomena. Lastly, we prove that Furstenberg systems of generalbounded multiplicative functions have divisible spectrum. When these systemsare obtained using logarithmic averages, we show that trivial rational spectrumimplies a strong dilation invariance property, called strong stationarity, but,quite surprisingly, this property fails when the systems are obtained usingCes\`aro averages.

Furstenberg systems of pretentious and MRT multiplicative functions