Value distribution of multiplicative functions along linear fractional sequences

For $a,c\in \mathbb{N}$ and $b,d\in \mathbb{Z}$ such that the (non-empty) set $$R_{a,b,c,d}:=\{\frac{an+b}{cn+d}:n\in \mathbb{N}\}\cap ({\mathbb{Q}}_{>0}\setminus \{1\})$$ is multiplicatively recurrent, we give a complete characterization of the set of limit points of every unimodular multiplicative function $f\in \mathcal{M}$ along $R_{a,b,c,d}\mathrm{.}$ We show that the possible limit sets are either the finite subgroups of the unit circle or the entire circle, thereby extending the dichotomy of Klurman–Mangerel.