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

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