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

Alexandre Roy

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arXiv · 2610.11545

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 $\mathbb{C}$ 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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Alexandre Roy. 2026-10-08. Continuity of the Julia sets in non-archimedean and hybrid settings. https://arxiv.org/abs/2610.11545

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