We define the stable and unstable manifolds associated to a hyperbolic fixed point $u$ of a dynamical system $f \colon\Omega \to E$ on a normed vector space. These come in two flavours: local and global. The global (un)stable manifolds are easiest to define:
$$W^s(u,f) := \left\{v \in \Omega \mid \lim_{k \to \infty} f^k(v) = u \right\},$$
and
$$W^u(u,f) := \left\{v \in \Omega \mid \lim_{k \to \infty}f^{-k}(v) = u \right\},$$
i.e. the vectors which are positively (negatively) asymptotic to $u$ under $f$.

The local (un)stable manifolds, denoted by
$$W^s_{ \operatorname{loc},r}(u,f) \qquad \text{and} \qquad W^u_{ \operatorname{loc},r}(u,f)$$
respectively[1], consist of those vectors in the global (un)stable manifold that remain within the ball of radius $r$ about $u$ under all positive (negative) interations of $f$.

The name “(un)stable manifold” is very suggestive, and indeed these spaces are submanifolds of $E$. More precisely, for $r$ sufficiently small, the local (un)stable manifolds are embedded submanifolds diffeomorphic to disks in the (un)stable space.

The global (un)stable manifolds are slightly less well-behaved—they are merely immersed submanifolds.

The proofs of these assertions—which are collectively known as the Stable Manifold Theorem—will be presented over the next few lectures.

1. Messiest notation ever, I know... ↩︎