stable manifold

Let $X$ be a topological space, and $f:X\to X$ ahomeomorphism. If $p$ is a fixed point for $f$, the stable andunstable sets of $p$ are defined by

$W^s(f,p)$$=\{q\in X:f^n(q)\underset{n\to \mathrm{\infty }}{\overset{}{\to }}p\},$

$W^u(f,p)$$=\{q\in X:f^{-n}(q)\underset{n\to \mathrm{\infty }}{\overset{}{\to }}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

$W^s(f,p)$$=W^s(f^k,p)$

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

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

$W_{\text{loc}}^s(f,p,U)$$=\{q\in U:f^n(q)\in U\text{ for each }n\ge 0\},$

$W_{\text{loc}}^u(f,p,U)$$=W_{\text{loc}}^s(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))\underset{n\to \mathrm{\infty }}{\overset{}{\to }}0\},$

$W^u(f,p)$$=W^s(f^{-1},p),$

where $d$ is a metric for $X$. This definition clearlycoincides 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 $x\to \mathrm{\infty }$ or $-\mathrm{\infty }$, respectively.

Suppose now that $X$ is a compact smooth manifold, and $f$ is a ${𝒞}^k$diffeomorphism, $k\ge 1$. 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 ${𝒞}^k$ 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 ${𝒞}^k$ topology of ${\mathrm{Diff}}^k(X)$ (the space of all ${𝒞}^k$ diffeomorphisms from $X$ to itself). Finally, the stable and unstable sets are ${𝒞}^k$ 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).