Cubic rational expressions over a finite field

We study and partially classify cubic rational expressions over a finite field [Formula: see text], up to equivalence given by composition with independent Möbius transformations on each side. When [Formula: see text] is even we obtain a full classification. When [Formula: see text] is odd we restrict our classification to expressions with at most three ramification points. However, we prove a general upper bound of [Formula: see text] for the total number of equivalence classes.