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Ergodic exponential maps with escaping singular behaviours

Weiwei Cui, Jun Wang

TL;DR

This work investigates ergodicity in the escaping exponential family $f_{\lambda}(z)=\lambda e^z$ by constructing a parameter for which the singular value escapes to infinity yet the map is ergodic. It develops a perturbation-based construction around post-singularly finite maps, leveraging holomorphic motions and a blowing-up property to produce an escaping limiting parameter with slowly growing real parts of the singular orbit. The main result contradicts the intuition that faster escape enforces non-ergodicity and shows ergodic escaping maps are dense in the bifurcation locus $\mathcal{B}$. The techniques shed light on the structure of the exponential parameter space and provide a framework for studying ergodicity in transcendental dynamics via perturbations and detailed singular-orbit control.

Abstract

We construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that $e^z$ is not ergodic.

Ergodic exponential maps with escaping singular behaviours

TL;DR

This work investigates ergodicity in the escaping exponential family by constructing a parameter for which the singular value escapes to infinity yet the map is ergodic. It develops a perturbation-based construction around post-singularly finite maps, leveraging holomorphic motions and a blowing-up property to produce an escaping limiting parameter with slowly growing real parts of the singular orbit. The main result contradicts the intuition that faster escape enforces non-ergodicity and shows ergodic escaping maps are dense in the bifurcation locus . The techniques shed light on the structure of the exponential parameter space and provide a framework for studying ergodicity in transcendental dynamics via perturbations and detailed singular-orbit control.

Abstract

We construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that is not ergodic.
Paper Structure (4 sections, 8 theorems, 16 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 16 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Construction
  3. Some lemmas
  4. The construction and proof of ergodicity

Key Result

Theorem 1.1

There exists an escaping parameter $\lambda$ such that $f_{\lambda}$ is ergodic.

Figures (1)

  • Figure 1: Black ones denotes the singular orbit of the starting map $f_{\lambda_0}$. The blue ones illustrates singular orbit of the desired special post-singularly fixed parameter.

Theorems & Definitions (14)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.1
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 4 more