Meromorphic functions whose action on their Julia sets is Non-Ergodic

TL;DR

This paper characterizes when Nevanlinna functions exhibit non-ergodic behavior on their Julia sets, especially when all asymptotic values tend to infinity, resulting in the Julia set being the entire sphere.

Contribution

It completes the classification of ergodic versus non-ergodic actions of Nevanlinna functions on their Julia sets based on asymptotic value behavior.

Findings

If all asymptotic values land on infinity, the Julia set is the entire sphere.

Under these conditions, the function's action on the Julia set is non-ergodic.

Previous results covered cases with compact repellers or some asymptotic values at infinity.

Abstract

Nevanlinna functions are meromorphic functions with a finite number of asymptotic values and no critical values. In [KK2] it was 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. In [CJK4] we 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.