stable manifoldStableManifold2003-06-13 14:49:292006-06-28 04:52:19Definition% this is the default PlanetMath preamble. as your knowledge
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Let $X$ be a topological space, and $f\colon X\rightarrow X$ a
homeomorphism. If $p$ is a fixed point for $f$, the \emph{stable and
unstable sets} of $p$ are defined by
\begin{align*}
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\},
\end{align*}
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
\begin{align*}
W^s(f,p)&= W^s(f^k,p)\\
W^u(f,p)&=W^u(f^k,p).
\end{align*}
Given a neighborhood $U$ of $p$, the \emph{local stable and unstable sets} of $p$ are defined by
\begin{align*}
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).
\end{align*}
If $X$ is metrizable, we can define the stable and unstable sets for any point by
\begin{align*}
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),
\end{align*}
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 $\Cdiff^k$
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 all $\Cdiff^k$ diffeomorphisms from $X$ to itself). 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).