Meromorphic functions whose action on their Julia sets is non-ergodic

Nevanlinna functions are meromorphic functions with a finite number of asymptotic values and no critical values. Keen and Kotus (1995) proved that if the orbits of all the asymptotic values accumulate on a compact set on which the function acts as a repeller, then the function acts ergodically on its Julia set. Chen et al. (2024) proved the action of the function on its Julia set is still ergodic if some, but not all of the asymptotic values land on infinity, and the remaining ones land on a compact repeller. In this paper, we complete the characterization of ergodicity for Nevanlinna functions by proving that if all the asymptotic values land on infinity, then the Julia set is the whole sphere and the action of the map there is non-ergodic.