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PreprintCC BY 4.0arXiv

Continuity of the Julia sets in non-archimedean and hybrid settings

  • Alexandre Roy
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Abstract

Let k be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over k and over ℂ equipped with the hybrid norm. In both settings, we show an analogue of Mané--Sad--Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.

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Cite

BibTeXRIS
BibTeX
@misc{roy2026continuity,
  title = {{Continuity of the Julia sets in non-archimedean and hybrid settings}},
  author = {Alexandre Roy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11545},
  eprint = {2610.11545},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11545},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Continuity of the Julia sets in non-archimedean and hybrid settings
AU  - Alexandre Roy
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11545
UR  - https://arxiv.org/abs/2610.11545
AB  - Let k be a complete, non-archimedean. We study the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over k and over ℂ equipped with the hybrid norm. In both settings, we show an analogue of Mané--Sad--Sullivan's theorem about the continuity of the Julia set (in the Hausdorff topology). The non-archimedean case relies on the maximum modulus principle, which we show for analytic functions over analytic spaces without boundary.
ER  -