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

Tao Chen, Yunping Jiang, Linda Keen

TL;DR

The article completes the ergodicity classification for Nevanlinna meromorphic functions by proving that if all asymptotic values land on infinity, the Julia set is the Riemann sphere, and the action is non-ergodic. It achieves this through a rigorous asymptotic-tract analysis near infinity, leveraging Schwarzian theory, Liouville-type changes of variables, and a near-Markov partition to construct two disjoint wandering sets of positive Lebesgue measure. Consequently, for almost every $z$, the omega-limit set satisfies $\omega(z)=P_f$ and there is no invariant finite measure absolutely continuous with respect to Lebesgue measure, establishing a full non-ergodicity result in the sphere-Julia case. The work thus completes the ergodicity dichotomy for Nevanlinna functions, complementing prior results for mixed asymptotic-value dynamics and clarifying the global measure-theoretic behavior of these maps.

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.

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

TL;DR

The article completes the ergodicity classification for Nevanlinna meromorphic functions by proving that if all asymptotic values land on infinity, the Julia set is the Riemann sphere, and the action is non-ergodic. It achieves this through a rigorous asymptotic-tract analysis near infinity, leveraging Schwarzian theory, Liouville-type changes of variables, and a near-Markov partition to construct two disjoint wandering sets of positive Lebesgue measure. Consequently, for almost every , the omega-limit set satisfies and there is no invariant finite measure absolutely continuous with respect to Lebesgue measure, establishing a full non-ergodicity result in the sphere-Julia case. The work thus completes the ergodicity dichotomy for Nevanlinna functions, complementing prior results for mixed asymptotic-value dynamics and clarifying the global measure-theoretic behavior of these maps.

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.
Paper Structure (8 sections, 12 theorems, 117 equations, 6 figures)

This paper contains 8 sections, 12 theorems, 117 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Definitions and standard theorems
  4. Nevanlinna functions
  5. The behavior of $f$ in a neighborhood of infinity
  6. Basic calculations
  7. Disjoint wandering sets of positive Lebesgue measure
  8. Main Theorems 2 and 3

Key Result

Theorem 2.1

Let $f: D(z_0, r)\to \mathbb{C}$ be a univalent function, then for any $\eta<1$,

Figures (6)

  • Figure 1: The critical lines and sectors for $N=4$
  • Figure 2: The decomposition of $h_{i,U} \circ E_{i,U}$ as a map from the auxiliary plane to the dynamic plane
  • Figure 3: The map of the asymptotic tract $T_i$ (green) and its image under $f^{k_i+1}$ (red)
  • Figure 4: $Hor^i_k$ for $N=4$
  • Figure 5: Rays and curves in lemma \ref{['largek']}
  • ...and 1 more figures

Theorems & Definitions (25)

  • Theorem 2.1: Koebe Distortion Theorem
  • Theorem 2.2
  • Definition 1
  • Theorem 2.3
  • Theorem
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Remark 2.1
  • ...and 15 more