1) The particular case $\mathrm{X}=\mathrm{P}^{n}$ generalizes Theorem 6 in [42] according to which canonical metrics are locally constant on the classical Fatou set (meaning that the norm of a non-vanishing local section is locally constant). Indeed, the restriction to the set of smooth rigid points of a strongly harmonic function is locally constant. This follows from the fact that any such point has an affinoid neighbourhood $\mathrm{U}$ which is a polydisk, so that the absolute value of any invertible function on $\mathrm{U}$, hence any harmonic function on $\mathrm{U}$ is constant.

2) In the case $\mathrm{X}=\mathrm{P}^{1}$, Fatou and Julia sets in the Berkovich framework have been studied by Rivera-Letelier [48] and Benedetto [9] ; see also [6] for a detailed exposition of the theory and further references. An example of Rivera-Letelier on the projective line (Example 10.70 of [6]) shows that the equicontinuity locus $\mathrm{E}_{f}$ may be smaller than the complement of the support of the measure $c_{1}(\overline{L})$.

Anyway, this proposition suggests the interest of a general study of Fatou sets and of pluripotential theory on Berkovich spaces. For example, is there an interesting theory of pseudoconvexity for Berkovich spaces ? Is it related to Stein spaces ? By analogy to the complex case (see [52]), are Berkovich Fatou components pseudoconvex ? Stein ?