Hyperbolic set

In

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In the special case when the entire manifold M is hyperbolic, the map f is called an Anosov diffeomorphism. The dynamics of f on a hyperbolic set, or hyperbolic dynamics, exhibits features of local structural stability and has been much studied, cf. Axiom A.

Definition

Let M be a

on M, the restriction of Df to E^s must be a contraction and the restriction of Df to E^u must be an expansion. Thus, there exist constants 0<λ<1 and c>0 such that

${\displaystyle T_{\Lambda }M=E^{s}\oplus E^{u}}$

and

${\displaystyle (Df)_{x}E_{x}^{s}=E_{f(x)}^{s}}$ and ${\displaystyle (Df)_{x}E_{x}^{u}=E_{f(x)}^{u}}$ for all ${\displaystyle x\in \Lambda }$

and

${\displaystyle \|Df^{n}v\|\leq c\lambda ^{n}\|v\|}$ for all ${\displaystyle v\in E^{s}}$ and ${\displaystyle n>0}$

and

${\displaystyle \|Df^{-n}v\|\leq c\lambda ^{n}\|v\|}$ for all ${\displaystyle v\in E^{u}}$ and ${\displaystyle n>0}$.

If Λ is hyperbolic then there exists a Riemannian metric for which c = 1 — such a metric is called adapted.

Examples

• Hyperbolic equilibrium point p is a fixed point, or equilibrium point, of f, such that (Df)_p has no eigenvalue with absolute value 1. In this case, Λ = {p}.
• More generally, a
  periodic orbit
  of f with period n is hyperbolic if and only if Dfⁿ at any point of the orbit has no eigenvalue with absolute value 1, and it is enough to check this condition at a single point of the orbit.

References

• Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. Reading Mass.: Benjamin/Cummings.
  ISBN 0-8053-0102-X
  .
• Brin, Michael; Stuck, Garrett (2002). Introduction to Dynamical Systems. Cambridge University Press.
  ISBN 0-521-80841-3
  .

This article incorporates material from Hyperbolic Set on

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