Minimality of p-adic rational maps with good reduction

2015/11/16 by Fan, Ai-Hua, Fan, Shilei, Liao, Lingmin +1 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1511.04856

Abstract

A rational map with good reduction in the field ℚ_p of p-adic numbers defines a 1-Lipschitz dynamical system on the projective line ℙ¹(ℚ_p) over ℚ_p. The dynamical structure of such a system is completely described by a minimal decomposition. That is to say, ℙ¹(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 ℙ¹(ℚ_2).

Cited by

• p-adic dynamical systems for p-adic λ-models on the Cayley trees

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