Minimality of p-adic rational maps with good reduction

Ai-Hua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang

Published 2015-11-16Version 1

A rational map with good reduction in the field $\mathbb{Q}\_p$ of $p$-adic numbers defines a $1$-Lipschitz dynamical system on the projective line $\mathbb{P}^1(\mathbb{Q}\_p)$ over $\mathbb{Q}\_p$. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, $\mathbb{P}^1(\mathbb{Q}\_p)$ is decomposed into three parts: finitely many periodic orbits; finite or countably many minimal subsystems each consisting of a finite union of balls; and the attracting basins of the periodic orbits and minimal subsystems. For any prime $p$, a criterion of minimality for rational maps with good reduction is obtained. When $p=2$, a complete characterization of minimal rational maps with good reduction is given in terms of their coefficients. It is also proved that a rational map of degree $2$ or $3$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}\_2)$.