PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (512)
Corrections (10)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
stable manifold (Definition)

Let $ X$ be a topological space, and $ f\colon X\rightarrow X$ a homeomorphism. If $ p$ is a fixed point for $ f$, the stable and unstable sets of $ p$ are defined by

$\displaystyle W^s(f,p)$ $\displaystyle =\{q\in X: f^n(q)\xrightarrow[n\rightarrow\infty]{} p\},$    
$\displaystyle W^u(f,p)$ $\displaystyle =\{q\in X: f^{-n}(q)\xrightarrow[n\rightarrow\infty]{} p\},$    

respectively.

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

$\displaystyle W^s(f,p)$ $\displaystyle = W^s(f^k,p)$    
$\displaystyle W^u(f,p)$ $\displaystyle =W^u(f^k,p).$    

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

$\displaystyle W^s_{\text{loc}}(f,p,U)$ $\displaystyle = \{q\in U: f^n(q)\in U \textnormal{ for each } n\geq 0\},$    
$\displaystyle W^u_{\text{loc}}(f,p,U)$ $\displaystyle = W^s_{\text{loc}}(f^{-1},p,U).$    

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

$\displaystyle W^s(f,p)$ $\displaystyle = \{q\in U: d(f^n(q),f^n(p))\xrightarrow[n\rightarrow\infty]{} 0\},$    
$\displaystyle W^u(f,p)$ $\displaystyle = W^s(f^{-1},p),$    

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 $ W^s(f,K)$ and $ W^u(f,K)$ (or just $ W^s(K)$ and $ W^u(K)$) the stable and unstable sets of $ K$, defined as the set of points $ x\in X$ such that $ d(f^n(x),K)\to 0$ when $ x\to \infty$ or $ -\infty$, respectively.

Suppose now that $ X$ is a compact smooth manifold, and $ f$ is a $ \mathcal{C}^k$ diffeomorphism, $ k\geq 1$. 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 $ \mathcal{C}^k$ embedded disks, whose tangent spaces at $ p$ are $ E^s$ and $ E^u$ (the stable and unstable spaces of $ Df(p)$), respectively; moreover, they vary continuously (in certain sense) in a neighborhood of $ f$ in the $ \mathcal{C}^k$ topology of $ \operatorname{Diff}^k(X)$ (the space of all $ \mathcal{C}^k$ diffeomorphisms from $ X$ to itself). Finally, the stable and unstable sets are $ \mathcal{C}^k$ 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).



"stable manifold" is owned by Koro.
(view preamble)

View style:

See Also: hyperbolic fixed point

Other names:  stable set, unstable set, unstable manifold
Log in to rate this entry.
(view current ratings)

Cross-references: hyperbolic set, topology, unstable, tangent spaces, stable manifold theorem, hyperbolic periodic point, diffeomorphism, smooth manifold, compact, invariant subset, metric, point, metrizable, neighborhood, least period, periodic point, fixed point, homeomorphism, topological space
There are 9 references to this entry.

This is version 8 of stable manifold, born on 2003-06-13, modified 2006-06-28.
Object id is 4357, canonical name is StableManifold.
Accessed 8558 times total.

Classification:
AMS MSC: 37C75 (Dynamical systems and ergodic theory :: Smooth dynamical systems: general theory :: Stability theory)
 37D10 (Dynamical systems and ergodic theory :: Dynamical systems with hyperbolic behavior :: Invariant manifold theory)
Pending Errata and Addenda
None.
[ View all 2 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)