Dynamics of transcendental entire maps on Berkovich affine line
In this article, the authors study the dynamics of transcendental entire maps on the Berkovich affine line \(\mathbb{A}_{\mathrm{Berk}}^1\) for an algebraically closed field \(K\) which is complete with respect to a non-trivial absolute non-Archimedean value \(|\cdot|\). For a transcendental entire map \(f: \mathbb{A}_{\mathrm{Berk}}^1 \to\mathbb{A}_{\mathrm{Berk}}^1\), the authors show that the Julia set \({\mathcal{J}}_f\) is the closure of the set of repelling periodic points of \(f\) in \(\mathbb{A}_{\mathrm{Berk}}^1\). Further, for a subset \(U\) of the Fatou set \({\mathcal{F}}_f\), the complement of \({\mathcal{J}}_f\), which is multiply connected, they also show that \(U\) is a wandering component, its image under the \(n\)th iterate of \(f\), \(f^n(U)\), is multiply connected, and for every \(x\in U\), \(\lim_{n\to \infty} f^n(x)\) diverges; moreover, if \(U\) is a finitely multiply connected component, it is also proved that there exists a positive integer \(n\) large enough such that \(f^n(U)\) is \(2\)-connected.
scientific article; zbMATH DE number 6160883