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

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.