When all asymptotic values of a Nevanlinna function land at infinity, its Julia set is the full sphere and the map acts non-ergodically there.
Zakeri, A Course in Complex Analysis , Princeton University Press, Princeton, NJ
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Meromorphic functions whose action on their Julia sets is Non-Ergodic
When all asymptotic values of a Nevanlinna function land at infinity, its Julia set is the full sphere and the map acts non-ergodically there.