# hyperbolic fixed point

Let $M$ be a smooth manifold. A fixed point $x$ of a diffeomorphism $f\colon M\to M$ is said to be a if $Df(x)$ is a linear hyperbolic isomorphism. If $x$ is a periodic point of least period $n$, it is called a hyperbolic periodic point if it is a hyperbolic fixed point of $f^{n}$ (the $n$-th iterate of $f$).

If the dimension of the stable manifold of a fixed point is zero, the point is called a source; if the dimension of its unstable manifold is zero, it is called a sink; and if both the stable and unstable manifold have nonzero dimension, it is called a saddle.

 Title hyperbolic fixed point Canonical name HyperbolicFixedPoint Date of creation 2013-03-22 13:47:57 Last modified on 2013-03-22 13:47:57 Owner Koro (127) Last modified by Koro (127) Numerical id 6 Author Koro (127) Entry type Definition Classification msc 37C25 Classification msc 37D05 Related topic StableManifold Related topic HyperbolicSet Defines hyperbolic periodic point Defines source Defines sink Defines saddle