On some classification problems of multiplicative functions

We prove that a multiplicative function $f:\mathbb{N}\to\mathbb{C}$ is Toeplitz if and only if there are a Dirichlet character $\chi$ and a finite subset $F$ of prime numbers such that $f(n)=\chi(n)$ for each $n$ which is coprime to all numbers from $F$. All such functions bounded by~1 are necessarily pretentious and they have exactly one Furstenberg system. Moreover, we characterize the class of pretentious functions that have precisely one Furstenberg system as those being Besicovitch (rationally) almost periodic. As a consequence, we show that the corrected Elliott's conjecture implies Frantzikinakis-Host's conjecture on the uniqueness of Furstenberg system for all real-valued bounded by~1 multiplicative functions. We also clarify relations between different classes of aperiodic multiplicative functions.