If is a periodic point of least period , then it is a fixed point of , and the stable and unstable sets of are
Given a neighborhood of , the local stable and unstable sets of are defined by
If is metrizable, we can define the stable and unstable sets for any point by
where is a metric for . This definition clearly coincides with the previous one when is a periodic point.
When is an invariant subset of , one usually denotes by and (or just and ) the stable and unstable sets of , defined as the set of points such that when or , respectively.
Suppose now that is a compact smooth manifold, and is a diffeomorphism, . If is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood of , the local stable and unstable sets are embedded disks, whose tangent spaces at are and (the stable and unstable spaces of ), respectively; moreover, they vary continuously (in certain sense) in a neighborhood of in the topology of (the space of all diffeomorphisms from to itself). Finally, the stable and unstable sets are injectively immersed disks. This is why they are commonly called stable and unstable manifolds. This result is also valid for nonperiodic points, as long as they lie in some hyperbolic set (stable manifold theorem for hyperbolic sets).
|Date of creation||2013-03-22 13:41:07|
|Last modified on||2013-03-22 13:41:07|
|Last modified by||Koro (127)|