Minimality of p-adic rational maps with good reduction

Aihua Fan; Shilei Fan; Lingmin Liao; Yuefei Wang

doi:10.3934/dcds.2017135

Article Contents

2017, Volume 37, Issue 6: 3161-3182. Doi: 10.3934/dcds.2017135

This issue Previous Article Dynamic rays of bounded-type transcendental self-maps of the punctured plane Next Article Existence of the solution for the viscous bipolar quantum hydrodynamic model

Minimality of p-adic rational maps with good reduction

1.
School of Mathematics and Statistics, Central China Normal University, 430079 Wuhan, China & LAMFA, UMR 7352, CNRS, Université de Picardie Jules Verne, 33, Rue Saint Leu, 80039 Amiens Cedex 1, France
2.
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China
3.
LAMA, UMR 8050, CNRS, Université Paris-Est Créteil, 61 Avenue du Général de Gaulle, 94010, Créteil Cedex, France
4.
Institute of Mathematics, AMSS, Chinese Academy of Sciences, 100190 Beijing, China

* Corresponding author: Shilei Fan
* Corresponding author: Shilei Fan

Received: February 29, 2016

Revised: January 31, 2017

Published: May 2017

A. H. FAN was supported by NSF of China (Grant No. 11471132); S. L. FAN was supported by NSF of China (Grant No. 11401236) and self-determined research funds of CCNU (Grant No. CCNU17QN0009); Y. F. WANG was supported by NSF of China (Grant No. 11231009)

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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 degrees $2$, $3$ and $4$ can never be minimal on the whole space $\mathbb{P}^1(\mathbb{Q}_2)$.

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Mathematics Subject Classification: Primary: 37P05; Secondary: 11S82, 37B05.

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Figure 1. Tree structure of $\mathbb{P}^1(\mathbb{Q}_2)$. The points of $\mathbb{P}^1(\mathbb{Q}_2)$ are considered as the boundary points of the infinite tree

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Figure 2. Tree structure of $\mathbb{P}^1(\mathbb{Q}_3)$. The points of $\mathbb{P}^1(\mathbb{Q}_3)$ are considered as the boundary points of the infinite tree.

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Coefficients $a_0, a_1, a_2, a_3, b_1, b_2, b_3$	Induced periodic orbits at level 3
1, 0, 1, 3, 3, 1, 0 1, 1, 1, 2, 3, 2, 3 1, 2, 1, 1, 3, 3, 2 1, 2, 3, 3, 3, 3, 0 1, 3, 1, 0, 3, 0, 1 1, 3, 3, 2, 3, 0, 3 3, 2, 1, 3, 1, 1, 0 3, 3, 1, 2, 1, 2, 3	$0\to \widetilde{0}\to 1\to 6 \to \widetilde{6}\to 3 $ $\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 7$
1, 0, 1, 3, 1, 1, 2 1, 1, 1, 2, 1, 2, 1 1, 2, 1, 1, 1, 3, 0 1, 2, 3, 3, 1, 3, 2 1, 3, 1, 0, 1, 0, 3 1, 3, 3, 2, 1, 0, 1 3, 2, 1, 3, 3, 1, 2 3, 3, 1, 2, 3, 2, 1	$0\to\widetilde{0}\to1\to6\to\widetilde{2}\to3$ $\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to7$
1, 0, 1, 3, 1, 1, 0 1, 1, 1, 2, 1, 2, 3 1, 2, 1, 1, 1, 3, 2 1, 2, 3, 3, 1, 3, 0 1, 3, 1, 0, 1, 0, 1 1, 3, 3, 2, 1, 0, 3 3, 2, 1, 3, 3, 1, 0 3, 3, 1, 2, 3, 2, 3	$0\to \widetilde{0}\to1\to2\to\widetilde{6}\to3$ $\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to7$
1, 0, 1, 3, 3, 1, 2 1, 1, 1, 2, 3, 2, 1 1, 2, 1, 1, 3, 3, 0 1, 2, 3, 3, 3, 3, 2 1, 3, 1, 0, 3, 0, 3 1, 3, 3, 2, 3, 0, 1 3, 2, 1, 3, 1, 1, 2 3, 3, 1, 2, 1, 2, 1	$0\to\widetilde{0}\to1\to2\to\widetilde{2}\to3$ $\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to7$
1, 0, 3, 1, 3, 1, 0 1, 1, 3, 0, 3, 2, 3 3, 0, 1, 1, 1, 3, 0 3, 0, 3, 3, 1, 3, 2 3, 1, 1, 0, 1, 0, 3 3, 1, 3, 2, 1, 0, 1 3, 2, 3, 1, 1, 1, 0 3, 3, 3, 0, 1, 2, 3	$0\to \widetilde{0}\to1\to 6\to \widetilde{6}\to 7$ $\to 4\to \widetilde{4}\to 5\to 2\to \widetilde{2}\to 3$
1, 0, 3, 1, 1, 1, 2 1, 1, 3, 0, 1, 2, 1 3, 0, 1, 1, 3, 3, 2 3, 0, 3, 3, 3, 3, 0 3, 1, 1, 0, 3, 0, 1 3, 1, 3, 2, 3, 0, 3 3, 2, 3, 1, 3, 1, 2 3, 3, 3, 0, 3, 2, 1	$0\to\widetilde{0}\to1\to6\to\widetilde{2}\to7$ $\to4\to\widetilde{4}\to5\to2\to\widetilde{6}\to3$
1, 0, 3, 1, 1, 1, 0 1, 1, 3, 0, 1, 2, 3 3, 0, 1, 1, 3, 3, 0 3, 0, 3, 3, 3, 3, 2 3, 1, 1, 0, 3, 0, 3 3, 1, 3, 2, 3, 0, 1 3, 2, 3, 1, 3, 1, 0 3, 3, 3, 0, 3, 2, 3	$0\to\widetilde{0}\to1\to2\to\widetilde{6}\to7$ $\to4\to\widetilde{4}\to5\to6\to\widetilde{2}\to3$
1, 0, 3, 1, 3, 1, 2 1, 1, 3, 0, 3, 2, 1 3, 0, 1, 1, 1, 3, 2 3, 0, 3, 3, 1, 3, 0 3, 1, 1, 0, 1, 0, 1 3, 1, 3, 2, 1, 0, 3 3, 2, 3, 1, 1, 1, 2 3, 3, 3, 0, 1, 2, 1	$0\to\widetilde{0}\to1\to2\to\widetilde{2}\to7$ $\to4\to\widetilde{4}\to5\to6\to\widetilde{6}\to3$

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