  The M\"obius disjointness conjecture of Sarnak states that the M\"obius
function does not correlate with any bounded sequence of complex numbers
arising from a topological dynamical system with zero topological entropy. We
verify the logarithmically averaged variant of this conjecture for a large
class of systems, which includes all uniquely ergodic systems with zero
entropy. One consequence of our results is that the Liouville function has
super-linear block growth. Our proof uses a disjointness argument and the key
ingredient is a structural result for measure preserving systems naturally
associated with the M\"obius and the Liouville function. We prove that such
systems have no irrational spectrum and their building blocks are infinite-step
nilsystems and Bernoulli systems. To establish this structural result we make a
connection with a problem of purely ergodic nature via some identities recently
obtained by Tao. In addition to an ergodic structural result of Host and Kra,
our analysis is guided by the notion of strong stationarity which was
introduced by Furstenberg and Katznelson in the early 90's and naturally plays
a central role in the structural analysis of measure preserving systems
associated with multiplicative functions.
