Gluing Dynamics: $\varepsilon$-Precision in Solving a Non-Archimedean Inverse Problem

This research proposes a new method for approximating the solution of the inverse problem of finding a rational function that generates known local dynamics within distinct, disjoint closed balls in non-Archimedean fields. Although our approach is not directly influenced by Runge's theorem for approximating analytic maps in complex settings, it shares similarities by adapting these ideas to the non-Archimedean context. We aim to connect local dynamic behaviors, similar to dynamic surgery, without using quasiconformal but rational mappings. Our main theorem and corollaries present an algorithmic technique to construct a rational function, denoted as $F_\varepsilon$, that synthesizes specified local dynamics with an $\varepsilon$-precision parameter globally.