shadowing lemma

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

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

单词	ShadowingLemma
释义	shadowing lemma Let $M$ be a Riemannian manifold, $f\\colon M\\to M$ a diffeomorphism and $\\Lambda\\subset M$ a compact hyperbolic set for $f$ . Then there is a neighborhood $U$ of $\\Lambda$ such that for every $\\delta>0$ there is an $\\epsilon>0$ so that every $\\epsilon$ -orbit (http://planetmath.org/PseudoOrbit) in $U$ is $\\delta$ -shadowed by an orbit of $f$ . Moreover, there is $\\delta_{0}>0$ such that, if $\\delta<\\delta_{0}$ and if the pseudo-orbit is bi-infinite,then the shadowing orbit is unique; and if $\\Lambda$ has a local product structure then the shadowing orbit is in $\\Lambda$ .
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