Ergodicity in some families of Nevanlinna Functions

TL;DR

This paper investigates ergodic properties of certain transcendental Nevanlinna functions with mixed asymptotic value behaviors, extending previous results to cases where some asymptotic values are repelling and others are prepoles.

Contribution

It introduces the first analysis of ergodicity in Nevanlinna functions with a mix of asymptotic value types, broadening understanding of their dynamical behavior.

Findings

Ergodicity holds when some asymptotic values are repellers and others are prepoles.

Extends previous ergodicity results to mixed asymptotic value cases.

Provides new insights into the dynamics of transcendental meromorphic functions.

Abstract

We study Nevanlinna functions f that are transcendental meromorphic functions having N asymptotic values and no critical values. In [KK] it was 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 this 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.