MINIMALITY OF p-ADIC RATIONAL MAPS WITH GOOD REDUCTION

Ai-Hua Fan (1) , Shilei Fan (2) , Lingmin Liao (3) , Yuefei Wang (4)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 Central China Normal University [Wuhan, China]
3 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
4 AMSS - Academy of Mathematics and Systems Science

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 $\mathbb{P}^1(\mathbb{Q}_2)$.

Mots clés: p-adic dynamical system; minimal decomposition; rational map; projective line; good reduction

Domaines: Systèmes dynamiques [math.DS]

Ai-Hua Fan, Shilei Fan, Lingmin Liao, Yuefei Wang. MINIMALITY OF p-ADIC RATIONAL MAPS WITH GOOD REDUCTION. Discrete and Continuous Dynamical Systems - Series A, 2017. ⟨hal-01227597v3⟩