Reaching fixed points as limits in subspaces
type: Item
instance of: scholarly article
title: Reaching fixed points as limits in subspaces
zbMATH Open document ID: 1359.54026
author: Sibylla Priess-Crampe
published in: Journal of Geometry
publication date: 2017-02-02T00:00:00Z
review text: Let \(X\) be a non-empty set, and \(\Gamma\) a partially ordered set with the smallest element \(0\). A mapping \(d:X\times X\to\Gamma\) is said to be an ultrametric distance if it satisfies the following three conditions for all \(x,y,z\in X\) and \(\gamma\in\Gamma\): (D1) \(d(x,y)=0\) if and only if \(x=y\). (D2) \(d(x,y)=d(y,x)\). (D3) If \(d(x,y)\leq\gamma\) and \(d(y,z)\leq\gamma\), then \(d(x,z)\leq\gamma\). The triple \((X,d,\Gamma)\) is called an ultrametric space. In this paper the author considers ultrametric spaces with a totally ordered value set \(\Gamma\). In this case, (D3) takes the simpler form (D3') \(d(x,z)\leq\max\{d(x,y),d(y,z)\}\). The author proves that a strictly contracting mapping of a spherically complete ultrametric space has a unique fixed point. Under certain assumptions, the author finds out that it is possible to cut the pseudo-convergent family in such a way that in an appropriate subspace the initial part of the family is a Cauchy family which has the fixed point as its limit, and then studies the application of the results to valued fields. Results by Kaplansky and a generalization of his crucial lemma for the uniqueness of maximal immediate extensions are used to prove that any subfield of a maximal valued field \(K\) has a spherical completion in \(K\). Finally, the results are applied to approximate roots of polynomials that satisfy Hensel's Lemma.
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 6680054
zbMATH Keywords: ultrametric spaces
zbMATH Keywords: valued fields
zbMATH Keywords: approximation of fixed points
reviewed by: Hüseyin Çakallı
MaRDI profile type: Publication
full work available at URL: s00022-016-0313-7
OpenAlex ID: W2276322007
cites work: Maximal fields with valuations
cites work: Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields
cites work: Approximation of elements in Henselizations
cites work: Untersuchungen zur arithmetischen Theorie der Körper. (Die Theorie der Teilbarkeit in allgemeinen Körpern. I-III.)
cites work: The Banach fixed point theorem for ultrametric spaces
cites work: Generalized ultrametric spaces. II
cites work: Fixed point and attractor theorems for ultrametric spaces
cites work: Differential equations over valued fields (and more)
cites work: Ultrametric dynamics
cites work: The approximation to a fixed point
cites work: Ultrametric spaces and logic programming
DOI: 10.1007/S00022-016-0313-7
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description: scientific article; zbMATH DE number 6680054