# 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 $\u03f5>0$ so that every $\u03f5$-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 |