SEMIGROUP OF RATIONAL $p$-ADIC FUNCTIONS FOR COMPOSITION

Abstract

We are interested in the Julia set of a semigroup of rational functions with coefficients in $\mathbb{C}_p$ where the semigroup operation is composition. We prove that if a semigroup $G$ is generated by a finite number of rational functions of degree at least two with coefficients in a finite extension of $\mathbb{Q}_p$, and has a nonempty Julia set $\mathcal{J}(G)$, then $\mathcal{J}(G)$ is perfect and has an empty interior.