Ergodicity in some families of Nevanlinna functions

We study Nevanlinna functions f, that is, transcendental meromorphic functions having N asymptotic values and no critical values. Keen and Kotus (1999) proved that if the orbits of all the asymptotic values have accumulation sets that are compact and on which f is a repeller, then f acts ergodically on its Julia set. In the present paper we prove that if some but not all of the asymptotic values have this property, while the others are prepoles, the same holds true. This is the first paper to consider this mixed case.