Hyperbolic set

In mathematics, a subset of a manifold is said to have hyperbolic structure with rspect to a map "f", when its tangent bundle may be split into two invariant subbundles, one of which is contracting, and the other expanding with respect to "f".

Definition

Let "M" be a compact smooth manifold, and let $f:M\; o\; M$ be a diffeomorphism. An "f"-invariant subset $Lambda$ of "M" is said to be hyperbolic (or to have a hyperbolic structure) if there is a splitting of the tangent bundle of "M" restricted to $Lambda$ into a Whitney sum of two $Df$-invariant subbundles, $E^s$ and $E^u$, the stable bundle and the unstable bundle. The splitting is such that the restriction of $Df|\_\{E^s\}$ is a contraction and $Df|\_\{E^u\}$ is an expansion. This means that there are constants

:$T\_Lambda\; M\; =\; E^soplus\; E^u$

and

:$Df(x)E^s\_x\; =\; E^s\_\{f(x)\}$ and $Df(x)E^u\_x\; =\; E^u\_\{f(x)\}$ for each $xin\; Lambda$

and

:$|Df^nv|\; le\; clambda^n|v|$ for each $vin\; E^s$ and $n>\; 0$

and

:$|Df^\{-n\}v|\; le\; clambda^n\; |v|$ for each $vin\; E^u$ and $n>0$.

using some Riemannian metric on "M". If $Lambda$ is hyperbolic, then there exists an adapted Riemannian metric, that is, one such that "c"=1.

When the subset Λ is the entire manifold "M", then the diffeomorphism "f" is called an Anosov diffeomorphism.

ee also

* Hyperbolic fixed point

References

* Ralph Abraham and Jerrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin/Cummings Publishing, Reading Mass. ISBN 0-8053-0102-X