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单词 HyperbolicSet
释义

hyperbolic set


Let M be a compact smooth manifoldMathworldPlanetmath, and let f:MM be a diffeomorphism.An f-invariant subset Λ of M is said to be hyperbolic (or to have an hyperbolic structure) if there is a splitting of the tangent bundle of M restricted to Λ into a (Whitney) sum of two Df-invariant subbundles, Es and Eu such that the restriction of Df|Es is a contraction and Df|Eu is an expansion. This means that there are constants 0<λ<1 and c>0 such that

  1. 1.

    TΛM=EsEu;

  2. 2.

    Df(x)Exs=Ef(x)s and Df(x)Exu=Ef(x)u for each xΛ;

  3. 3.

    Dfnv<cλnv for each vEs and n>0;

  4. 4.

    Df-nv<cλnv for each vEu and n>0.

using some Riemannian metricMathworldPlanetmath on M.

If Λ is hyperbolic, then there exists an adapted Riemannian metric, i.e. one such that c=1.

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更新时间:2025/5/4 22:27:38