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Ergodicity in some families of Nevanlinna Functions

Tao Chen, Yunping Jiang, Linda Keen

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.

Ergodicity in some families of Nevanlinna 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.
Paper Structure (7 sections, 13 theorems, 74 equations)

This paper contains 7 sections, 13 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Main Theorem
  4. The measure of the set $I$
  5. Proof of Theorem \ref{['main1']}
  6. Proof of Theorem \ref{['lem2']}
  7. Proof of the main theorem

Key Result

Theorem 1

The Schwarzian derivative of a meromorphic function with finitely many critical points and finitely many asymptotic values is a rational function. If there are no critical points, it is a polynomial. Conversely, if a meromorphic function has a rational Schwarzian derivative, it has finitely many cri

Theorems & Definitions (28)

  • Remark 1.1
  • Conjecture 1
  • Definition 1
  • Theorem 1
  • Definition 2
  • Lemma 2
  • proof
  • Remark 2.1
  • Proposition 3
  • proof
  • ...and 18 more