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PreprintarXiv licencearXiv

Dynamical (non-)recurrence of escaping exponential maps

  • Weiwei Cui
  • Jiaxing Huang
  • Jun Wang
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Abstract

We show that exponential maps could have complicated measurable behaviours by giving two examples in the exponential family both with singular value escaping to infinity ``slowly", one of which is recurrent (in the sense that every positive measure set will return to itself infinitely many times), while the other is non-recurrent. This also complements earlier results of Lyubich, Rees, Urbański-Zdunik, and Hemke. We also give an example of exponential maps with singular value escaping to infinity arbitrarily slow, which is a parametric analog of a result of Rempe.

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Cite

BibTeXRIS
BibTeX
@misc{cui2026dynamical,
  title = {{Dynamical (non-)recurrence of escaping exponential maps}},
  author = {Weiwei Cui and Jiaxing Huang and Jun Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.03610},
  eprint = {2609.03610},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.03610},
}
RIS
TY  - UNPB
TI  - Dynamical (non-)recurrence of escaping exponential maps
AU  - Weiwei Cui
AU  - Jiaxing Huang
AU  - Jun Wang
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.03610
UR  - https://arxiv.org/abs/2609.03610
AB  - We show that exponential maps could have complicated measurable behaviours by giving two examples in the exponential family both with singular value escaping to infinity ``slowly", one of which is recurrent (in the sense that every positive measure set will return to itself infinitely many times), while the other is non-recurrent. This also complements earlier results of Lyubich, Rees, Urbański-Zdunik, and Hemke. We also give an example of exponential maps with singular value escaping to infinity arbitrarily slow, which is a parametric analog of a result of Rempe.
ER  -