Perturbed Dynamical Systems in $\mathfrak p$-Adic Fields

Let $k$ be a $\mathfrak p$-adic field, and let $\mathcal D$ be the class of all discrete dynamical systems defined by polynomials of the kind $h(x)=x+g(x)$, where $g(x)\in k[x]$ is irreducible. Using Krasner's lemma as a tool, we investigate the stability of this class with respect to perturbations of the kind $h_r(x)=h(x)+r(x)$, where $h(x)\in \mathcal D$ and $r(x)\in k[x]$.