Ultrametric dynamics
type: Item
instance of: scholarly article
title: Ultrametric dynamics
zbMATH Open document ID: 1262.54014
author: Sibylla Priess-Crampe
author: Paulo Ribenboim
published in: Illinois Journal of Mathematics
publication date: 2013-01-04T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1355927037
review text: The ultrametric \(d\) on a nonempty set \(X\) is a map \(a:X\times X\to \Gamma\) with values in an ordered set \((T_0\leq)\) with the smallest element \(0\), satisfying (1) \(d(x,y)= 0\quad\text{iff }x= y,\) (2) \(d(x,y)= d(y,x),\) (3) \(d(x,y)\leq\gamma\) and \(d(y,z)\leq\gamma\) imply \(d(x,z)\leq\gamma\) for \(x,y,z\in X\), \(\gamma\in\Gamma\). For any \(\gamma\in\Gamma\) and \(x\in X\) the ball \(B_\gamma(x)\) is defined as the set \(\{y\in X\mid d(y,x)\leq \gamma\}\). If there is a point \(z\) with \(d(x,z)= \gamma\), the ball \(B_\gamma(x)\) is called principal. The space is said to be spherically complete if every totally ordered set of balls has nonempty intersection. When this condition holds only for principal balls then the space is called principally complete. The authors study the dynamics on ultrametric spaces generated by a selfmap \(\varphi: X\to X\) or maps \(\psi_1,\psi_2: X\to X'\), satisfying certain conditions. For the selfmap \(\varphi: X\to X\) on \((X,d)\) the following conditions appear in their main result \[ x\neq y \Rightarrow d(\varphi(x), \varphi(y))< d(x,y),\tag{1} \] \[ \varphi(x)\neq x\Rightarrow d(\varphi^2(x), \varphi(x))< d(\varphi(x), x).\tag{2} \] They show that if the space \((X,d)\) is principally complete and \(\varphi\) satisfies condition (1), then \(\varphi\) has a unique fixed point. If it satisfies condition (2), then \(\varphi\) has a fixed point.
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6122120
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 6122120