shadowing lemma

Let $M$ be a Riemannian manifold, $f:M\to M$ a diffeomorphism and $\mathrm{\Lambda }\subset M$ a compact hyperbolic set for $f$. Then there is a neighborhood $U$ of $\mathrm{\Lambda }$ such that for every $\delta >0$ there is an $ϵ>0$ so that every $ϵ$-orbit (http://planetmath.org/PseudoOrbit) in $U$ is $\delta$-shadowed by an orbit of $f$.

Moreover, there is ${\delta }_0>0$ such that, if and if the pseudo-orbit is bi-infinite, then the shadowing orbit is unique; and if $\mathrm{\Lambda }$ has a local product structure then the shadowing orbit is in $\mathrm{\Lambda }$.

Title	shadowing lemma
Canonical name	ShadowingLemma
Date of creation	2013-03-22 14:07:22
Last modified on	2013-03-22 14:07:22
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	6
Author	Koro (127)
Entry type	Theorem
Classification	msc 37C50