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.