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

TL;DR

This paper investigates a specific class of transcendental meromorphic functions with two asymptotic values and no critical points, demonstrating that their action on Julia sets is non-ergodic, contrasting previous ergodicity results.

Contribution

It establishes non-ergodicity of these functions on Julia sets, providing new insights into the dynamics of transcendental meromorphic functions with particular value configurations.

Findings

Functions with two asymptotic values and no critical points are non-ergodic on Julia sets.

Contrasts with previous results showing ergodicity in similar function classes.

Highlights the diversity of dynamical behaviors in transcendental meromorphic functions.

Abstract

We study transcendental meromorphic functions having two prepole asymptotic values and no critical values. We prove that these functions acting on their Julia sets are non-ergodic, which illustrates the antithesis of the Keen-Kotus result in [KK2] on the ergodicity of another subfamily of functions with two asymptotic values and no critical values.