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Fix an integer $\ell \ge 3$. Rikuna introduced a polynomial $r(x,t)$ defined over a function field $K\left(t\right)$ whose Galois group is cyclic of order $\ell$, where $K$ satisfies some mild hypotheses. In this paper we define the family of ${\left\{r_n(x,t)\right\}}_{n\ge 1}$ of degree ${\ell }^n$. The $r_n(x,t)$ are constructed iteratively from the $r(x,t)$. We compute the Galois groups of the $r_n(x,t)$ for odd $\ell$ over an arbitrary base field and give applications to arithmetic dynamical systems.