Dynamics of quadratic polynomials over local fields
type: Item
instance of: scholarly article
title: Dynamics of quadratic polynomials over local fields
zbMATH Open document ID: 1197.37144
author: Jean-Yves Briend
author: Hervé Perdry
author: Robert Benedetto
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/45199
review text: The authors determine the structure of the Julia set for quadratic polynomials over a local field. Let \(K\) be a field complete under a rank one valuation \(v\), let \(P(X)=X^2+bX+c\in K[X]\) and put \(\Delta=(b-1)^2-4c\). Moreover denote by \(C_v\) the completion of the algebraic closure of \(K\). It is shown that if for all \(\xi\in K\) one has \(v(P(\xi)-\xi)>1\), then the set \(J_P(K)=J_P(C_v)\cap P^1(K)\) of \(K\)-points of the Julia set and the filled Julia set \(JR_P(K)\) of \(P\) are both empty. If for some \(\xi\in K\) one has \(v(P(\xi)-\xi)\leq 1\), then there are two cases: if \(v(\Delta)\leq1\), then \(J_P(K)\) is empty and \(JR_P(K)\) is a closed ball of radius one and center \(\xi\), and otherwise \(J_P(K)=JR_P(K)\) is a non-empty compact set on which \(P\) is topologically conjugated to the one-sided shift on two symbols.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 37P40
Mathematics Subject Classification ID: 11S05
Mathematics Subject Classification ID: 37P05
Mathematics Subject Classification ID: 37P20
zbMATH DE Number: 5302777
zbMATH Keywords: quadratic polynomials
zbMATH Keywords: Julia sets
zbMATH Keywords: local fields
MaRDI profile type: Publication
OpenAlex ID: W2059210667
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DOI: 10.5802/JTNB.589
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description: scientific article; zbMATH DE number 5302777