Shadowing is generic
type: Item
instance of: scholarly article
title: Shadowing is generic
zbMATH Open document ID: 0973.54035
DOI: 10.1016/S0166-8641(98)00062-5
author: O. B. Plamenevskaya
author: Sergei Yu. Pilyugin
published in: Topology and its Applications
publication date: 2001-11-23T00:00:00Z
review text: Let \(M\) be a smooth compact manifold without boundary. Let \(r\) be a Riemannian metric on \(M\). Consider the space \(Z(M)\) of homeomorphisms \(M\to M\) equipped with the \(C^0\) topology induced by the metric \(\rho(\varphi,\psi)=\max_{x\in M} r(\varphi(x),\psi(y))\). A sequence \(x_k\in M\), \(k\in Z\), is called a \(d\)-pseudotrajectory for \(\varphi\in Z(M)\) if \(r(x_{k+1},\varphi(x_k))<d\), \(k\in Z\). We say that \(\varphi\) has the shadowing property if for any \(\varepsilon>0\) there is \(d>0\) such that for every \(d\)-pseudotrajectory \(\{x_k\}\) there exists a point \(x\in M\) such that \(r(x_k,\varphi^k(x))<\varepsilon\), \(k\in Z\). The shadowing property is a key tool in the studies of chaotic maps. It is shown here that a generic homeomorphism \(\varphi\in Z(M)\) has the shadowing property. Previously, this fact was only proved when dim\(M\leq 3\).
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37C50
zbMATH DE Number: 1367547
zbMATH Keywords: shadowing property
zbMATH Keywords: pseudotrajectory
MaRDI profile type: Publication
Recommended article: Generic Spectrahedral Shadows
Recommended article: The simplest shadowing
Recommended article: Shadowing is generic on various one-dimensional continua with a special geometric structure
Recommended article: Shadowing is generic -- a continuous map case
Recommended article: Shadowing is generic on dendrites
Recommended article: Sub-shadowings
Recommended article: On genericity of shadowing in one dimension
Recommended article: A unified approach to theories of shadowing
Recommended article: Generalizations of the shadow problem
description: scientific article; zbMATH DE number 1367547