stable manifold

Let X be a topological spaceMathworldPlanetmath, and f:XX a homeomorphismPlanetmathPlanetmath. If p is a fixed pointMathworldPlanetmathPlanetmath for f, the stable and unstable sets of p are defined by

Ws(f,p) ={qX:fn(q)np},
Wu(f,p) ={qX:f-n(q)np},


If p is a periodic point of least period k, then it is a fixed point of fk, and the stable and unstable sets of p are

Ws(f,p) =Ws(fk,p)
Wu(f,p) =Wu(fk,p).

Given a neighborhoodMathworldPlanetmathPlanetmath U of p, the local stable and unstable sets of p are defined by

Wlocs(f,p,U) ={qU:fn(q)U for each n0},
Wlocu(f,p,U) =Wlocs(f-1,p,U).

If X is metrizable, we can define the stable and unstable sets for any point by

Ws(f,p) ={qU:d(fn(q),fn(p))n0},
Wu(f,p) =Ws(f-1,p),

where d is a metric for X. This definition clearly coincides with the previous one when p is a periodic point.

When K is an invariant subset of X, one usually denotes by Ws(f,K) and Wu(f,K) (or just Ws(K) and Wu(K)) the stable and unstable sets of K, defined as the set of points xX such that d(fn(x),K)0 when x or -, respectively.

Suppose now that X is a compactPlanetmathPlanetmath smooth manifoldMathworldPlanetmath, and f is a 𝒞k diffeomorphism, k1. If p is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood U of p, the local stable and unstable sets are 𝒞k embedded disks, whose tangent spaces at p are Es and Eu (the stable and unstable spaces of Df(p)), respectively; moreover, they vary continuously (in certain sense) in a neighborhood of f in the 𝒞k topology of Diffk(X) (the space of all 𝒞k diffeomorphisms from X to itself). Finally, the stable and unstable sets are 𝒞k 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).

Title stable manifold
Canonical name StableManifold
Date of creation 2013-03-22 13:41:07
Last modified on 2013-03-22 13:41:07
Owner Koro (127)
Last modified by Koro (127)
Numerical id 11
Author Koro (127)
Entry type Definition
Classification msc 37D10
Classification msc 37C75
Synonym stable set
Synonym unstable set
Synonym unstable manifold
Related topic HyperbolicFixedPoint