Examples related to Ulam's fixed point problem
type: Item
instance of: scholarly article
title: Examples related to Ulam's fixed point problem
zbMATH Open document ID: 0787.54041
DOI: 10.12775/TMNA.1993.013
author: Włodzimierz Kuperberg
author: Piotr Minc
author: Coke S. Reed
author: Krystyna Kuperberg
published in: Topological Methods in Nonlinear Analysis
publication date: 1994-01-06T00:00:00Z
review text: The paper deals with several examples related to the following problem posed by \textit{S. Ulam}: given a continuum (connected compact metric space) \(M\), does there exist a constant \(\varepsilon > 0\) such that any \(\varepsilon\)-map \(f: M\to M\) (i.e. continuous map whose trajectories \(\{f^ n(x)\}^ \infty_{n=0}\), \(x\in M\), are bounded by \(\varepsilon\)), possess a fixed point? An example of a one-dimensional continuum such that for any \(\varepsilon\) there is a fixed point free \(\varepsilon\)-involution is given. Then it is shown that the Cartesian product of the circle and the Hilbert cube for any \(\varepsilon > 0\) admits a fixed point free \(\varepsilon\)-homeomorphism. At last an example of the rest point free dynamical system with trajectories uniformly bounded by arbitrary \(\varepsilon > 0\) defined on a three-dimensional manifold is provided.
reviewed by: Wojciech Kryszewski
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 482113
zbMATH Keywords: continuum
zbMATH Keywords: dynamical system
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 482113