pseudo-orbit

Let $(X,d)$ be a metric space, $f\colon X\to X$ a function, and let $\epsilon>0$ . An $\epsilon$ -pseudo-orbit for $f$ is a sequence $\{x_{n}:n\in\mathbb{Z},\,a<n<b\}$ , where $-\infty\leq a<b\leq\infty$ , such that $d(x_{n+1},f(x_{n}))<\epsilon$ for all $a<n<b$ . A periodic pseudo-orbit is an infinite pseudo-orbit $\{x_{n}\}$ such that there is some $p$ with $x_{n+p}=x_{n}$ for all n.

Given $\delta>0$ , the pseudo-orbit $\{x_{n}:a<n<b\}$ is said to be $\delta$ -shadowed by the orbit of $x$ , if $d(x_{n},f^{n}(x))<\delta$ for all $a<n<b$ .

Title	pseudo-orbit
Canonical name	Pseudoorbit
Date of creation	2013-03-22 14:07:19
Last modified on	2013-03-22 14:07:19
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	7
Author	Koro (127)
Entry type	Definition
Classification	msc 37C50
Synonym	pseudo orbit
Synonym	$\epsilon$ -pseudo-orbit
Synonym	$\epsilon$ -orbit
Synonym	$\epsilon$ -chain
Defines	shadow
Defines	shadowing
Defines	shadowed