Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set.

Definition

The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics. Similar notions apply for systems whose time evolution is given by a flow.

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

W^s(f,p) =\{q\in X: f^n(q)\rightarrow p \mbox{ as } n\rightarrow\infty \}

and the unstable set of $p$ is defined by

W^u(f,p) =\{q\in X: f^{-n}(q)\rightarrow p \mbox{ as } n\rightarrow\infty \}.

Here, $f^{-1}$ denotes the preimage of the function (and not its inverse). Thus $f^{-2}$ is the preimage of the preimage, and so on.

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)

and

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^s_{\mathrm{loc}}(f,p,U) = \{q\in U: f^n(q)\in U \mbox{ for each } n\geq 0\}

and

W^u_{\mathrm{loc}}(f,p,U) = W^s_{\mathrm{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))\rightarrow 0 \mbox { for } n\rightarrow\infty \}

and

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 $\mathcal{C}^k$ 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 a certain sense) in a neighborhood of $f$ in the $\mathcal{c}^k$ topology of $\mathrm{Diff}^k(X)$ (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).

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Definition

See also