Minimality of p-adic rational maps with good reduction

TL;DR

This paper characterizes the minimality of p-adic rational maps with good reduction on the projective line over Q_p, providing criteria and structural decomposition, especially focusing on the case p=2 and degrees 2-4.

Contribution

It introduces a criterion for minimality of rational maps with good reduction over Q_p and analyzes the case p=2, degrees 2-4, revealing limitations on minimality.

Findings

A complete description of the dynamical structure of such systems.

A criterion for minimality of rational maps with good reduction for any prime p.

Rational maps of degree 2, 3, and 4 over Q_2 cannot be minimal on the entire space.

Abstract

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 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 condition in terms of the coefficients of the rational map is proved to be necessary for the map being minimal and having good reduction, and sufficient for the map being minimal and $1$-Lipschitz. It is also proved that a rational map having good reduction of degree $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}\_2)$.