Shadowing in dynamical systems
type: Item
instance of: scholarly article
title: Shadowing in dynamical systems
zbMATH Open document ID: 0954.37014
published in: Lecture Notes in Mathematics
publication date: 1999-11-11T00:00:00Z
review text: The book contains a detailed and comprehensive presentation of the shadowing property of dynamical systems. While the results contained in the book are not new, the author succeeds in presenting them in a unified way. The discussion is enhanced by a fairly large collection of fundamental examples. The basic object of consideration is a homeomorphism \(\varphi\) of a metric space \((X,r)\). This homeomorphism is said to have the pseudo orbit tracing property on a subset \(Y\) of \((X,r)\) if for a given \(\varepsilon> 0\) there exists some \(d>0\) such that for every sequence \(\{x_k\} \subset Y\) with \(r(\Phi(x_k), x_{k+1})< d\) a point \(x\in X\) can be found with \(r(\Phi^k x,x_k)< \varepsilon\) for all \(k\in \mathbb{Z}\). Section 1 of the book introduces this and related properties of a homeomorphism. It is shown that a diffeomorphism \(\varphi\) of a manifold \(M\) has a pseudo orbit tracing property on a neighborhood of a compact invariant hyperbolic set. For this fact, two proofs are given which go back to Anosov and Bowen. Moreover the section contains a detailed discussion of an analogous property for family of \(C^1\)-maps of Banach spaces to which many results on shadowing can be reduced. In Section 2, shadowing is related to other well known properties of topological dynamical systems. A result of Walters is presented which states that an expansive diffeomorphism of a smooth closed manifold which has the pseudo orbit tracing property is structurally stable. On the other hand, every structurally stable diffeomorphism has the pseudo orbit tracing property. It is shown that structural stability is in fact equivalent to a certain shadowing property. Section 3 contains a large collection of specific examples which illustrate the results presented in sections 1 and 2. Among these is a complete discussion of 1-dimensional dynamical systems, linear maps and their induced action on the sphere, lattice systems and attractors for evolution systems. Section 4 is de…
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 37-02
zbMATH DE Number: 1361782
zbMATH Keywords: structural stability
zbMATH Keywords: hyperbolic sets
zbMATH Keywords: shadowing
zbMATH Keywords: pseudo orbit tracing property
zbMATH Keywords: numerical treatment
zbMATH Keywords: reference book
zbMATH Keywords: topological dynamical systems
author: Sergei Yu. Pilyugin
MaRDI profile type: Publication
DOI: 10.1007/BFB0093184
description: scientific article; zbMATH DE number 1361782