Lower and upper bounds for \(\omega\)-limit sets of nonexpansive maps
type: Item
instance of: scholarly article
title: Lower and upper bounds for \(\omega\)-limit sets of nonexpansive maps
zbMATH Open document ID: 1011.47037
DOI: 10.1016/S0019-3577(01)80025-2
author: Roger D. Nussbaum
author: Bas Lemmens
author: Sjoerd M. Verduyn Lunel
published in: Indagationes Mathematicae. New Series
publication date: 2002-06-23T00:00:00Z
review text: Let \((X,d)\) be a complete metric space, \(D_f\) a closed subset of \(X\) and let \(f\) map \(D_f\) info itself. For each \(x\in D_f\) the \(\omega\)-limit set, \(\omega(x)=\omega(x;f)\), is defined by: \[ \omega(x)=\Bigl\{y\in D_f: y=\lim_{i\to\infty}f^{k_i}(x) \text{for some sequence of integers } k_i\to\infty \Bigr\}. \] In the paper under review, the authors discuss the procedure to obtain a priori upper bounds on the cardinality of the \(\omega\)-limit of \(l_1\)-norm nonexpansive maps, based on the approach introduced by \textit{M. Misiurewicz} [Math. Gottingensis, Schriftenr. Sonderforschungsbereichs Geom. Anal. 45 (1987; Zbl 0657.46012)]. In the case when the dimension of the space is less than six, they give a sharper upper bound for the cardinality of \(\omega\)-limit sets of \(l_1\)-norm nonexpansive maps than the upper bound obtained by Misiurewicz. The authors also describe two different procedures to construct periodic orbits of \(l_1\)-norm nonexpansive maps. From these constructions it follows, in particular, that a lower bound for the largest possible minimal period of a periodic point of an \(l_1\)-norm nonexpansive map is given by \(3\cdot 2^{n-1}\) for \(n\geq 3\). The paper also contains some interesting examples and open questions.
reviewed by: Dariusz Bugajewski
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 1758184
zbMATH Keywords: \(\omega\)-limit set
zbMATH Keywords: \(l_1\)-norm nonexpansive maps
zbMATH Keywords: a priori upper bounds on cardinality
zbMATH Keywords: periodic orbits
zbMATH Keywords: minimal period
MaRDI profile type: Publication
cites work: Nonlinear models of diffusion on a finite space
cites work: Separating pairs of points of standard boxes
cites work: Asymptotic behavior of nonlinear contraction semigroups
cites work: Monotonic Subsequences
cites work: Integral rigid sets and periods of nonexpansive maps
cites work: Q4623270
cites work: Q3996635
cites work: Q3805269
cites work: Omega limit sets of nonexpansive maps: Finiteness and cardinality estimates
cites work: Estimates of the periodic points for nonexpansive operators
cites work: Q4336960
cites work: Admissible Arrays and a Nonlinear Generalization of Perron-Frobenius Theory
cites work: Periodic points of nonexpansive maps and nonlinear generalizations of the Perron-Frobenius theory
cites work: Generalizations of the Perron-Frobenius theorem for nonlinear maps
cites work: Periods of nonexpansive operators on finite \(\ell _ 1\)-spaces
cites work: Erratum: Periods of non-expansive operators on finite \(\ell_ 1\)-spaces
cites work: Q3814277
full work available at URL: s0019-3577(01)80025-2
OpenAlex ID: W2166243027
Recommended article: A Nonlinear Perron-Frobenius Theorem
Recommended article: Integral rigid sets and periods of nonexpansive maps
Recommended article: Q4466588
Recommended article: A characterization of the periods of periodic points of 1-norm nonexpansive maps
Recommended article: Periods of periodic points of 1-norm nonexpansive maps
Recommended article: Q4026298
Recommended article: Dynamics of non-expansive maps on strictly convex Banach spaces
Recommended article: Omega limit sets of nonexpansive maps: Finiteness and cardinality estimates
Recommended article: Q4395220
Recommended article: Q4304365
description: scientific article; zbMATH DE number 1758184