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A counterexample to Eremenko's conjecture

  • David Martí-Pete
  • Lasse Rempe
  • James Waterman
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Abstract

The escaping set of a transcendental entire function consists of those points that tend to infinity under iteration. Eremenko asked whether every connected component of this set is unbounded. In arXiv:2108.10256, we constructed transcendental entire functions for which the escaping set has prescribed compact connected components; in particular, it may have a singleton component. This disproves the conjecture. In this note, we give an introduction to the problem, aimed at a general mathematical audience. We also sketch a variant of the construction in which a prescribed half-strip is an oscillating wandering domain and an escaping point lies on its boundary. The account is based on the second author's lecture at the 2026 International Congress of Basic Science.

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Cite

BibTeXRIS
BibTeX
@misc{martipete2026counterexample,
  title = {{A counterexample to Eremenko's conjecture}},
  author = {David Martí-Pete and Lasse Rempe and James Waterman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21977},
  eprint = {2609.21977},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21977},
}
RIS
TY  - UNPB
TI  - A counterexample to Eremenko's conjecture
AU  - David Martí-Pete
AU  - Lasse Rempe
AU  - James Waterman
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21977
UR  - https://arxiv.org/abs/2609.21977
AB  - The escaping set of a transcendental entire function consists of those points that tend to infinity under iteration. Eremenko asked whether every connected component of this set is unbounded. In arXiv:2108.10256, we constructed transcendental entire functions for which the escaping set has prescribed compact connected components; in particular, it may have a singleton component. This disproves the conjecture. In this note, we give an introduction to the problem, aimed at a general mathematical audience. We also sketch a variant of the construction in which a prescribed half-strip is an oscillating wandering domain and an escaping point lies on its boundary. The account is based on the second author's lecture at the 2026 International Congress of Basic Science.
ER  -