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| Title of object: |
stable manifold |
| Canonical Name: |
StableManifold |
| Type: |
Definition |
| Created on: |
2003-06-13 14:49:29 |
| Modified on: |
2003-06-13 14:58:06 |
| Classification: |
msc:37C75, msc:37D10 |
| Synonyms: |
stable manifold=stable set
stable manifold=unstable set
stable manifold=unstable manifold |
Preamble:
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Content:
Let $X$ be a topological space, and $f:X\rightarrow X$ a
homeomorphism. If $p$ is a fixed point for $f$, the \emph{stable and
unstable sets} of $p$ are defined by
$$W^s(f,p)=\{q\in X: f^n(q)\xrightarrow[n\rightarrow\infty]{} p\},$$
$$W^u(f,p)=\{q\in X: f^{-n}(q)\xrightarrow[n\rightarrow\infty]{} p\},$$ respectively.
If $p$ is a periodic point of least period $k$, then it is a fixed point of $f^k$, and the stable and ustable sets of $p$ are
$$W^s(f,p)= W^s(f^k,p)\textnormal{ and } W^u(f,p)=W^u(f^k,p).$$
Given a neighborhood $U$ of $p$, the \emph{local stable and ustable sets} of $p$ are defined by
$$W^s_{\text{loc}}(f,p,U) = \{q\in U: f^n(q)\in U \textnormal{ for each }
n\geq 0\},$$ $$W^u_{\text{loc}}(f,p,U) = W^s_{\text{loc}}(f^{-1},p,U).$$
If $X$ is metrizable, we can define the stable and unstable sets for any point by
$$W^s(f,p) = \{q\in U: d(f^n(q),f^n(p))\xrightarrow[n\rightarrow\infty]{} 0\},$$
$$W^u(f,p) = W^s(f^{-1},p),$$
where $d$ is a metric for $X$. This definition clearly
coincides with the previous one when $p$ is a periodic point.
Suppose now that $X$ is a compact smooth manifold, and $f$ is a $\Cdiff^k$
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 $\Cdiff^k$ 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 $\Cdiff^k$ topology of $\Diff^k(X)$ (the space of diffeomorphisms from $X$ to $X$). Finally, the stable and unstable sets are $\Cdiff^k$ injectively immersed disks. This is why they are commonly called \emph{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). |
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