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 $\C_p$ 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 $\Q_p$, and has a nonempty Julia set $\cj(G)$, then $\cj(G)$ is perfect and has an empty interior.