In this lecture we return to the stable manifolds from Lectures 33 and 34, only now in the setting of hyperbolic sets.

Let $f \colon M \to M$ be a dynamical system, and suppose $\Lambda \subseteq M$ is a compact hyperbolic set. Let $\widehat{f}$ denote the lifting of $f$ (as defined in the last lecture). We define two flavours of stable manifolds:

1. The local fibre stable manifold of $\widehat{f}$ at $x \in \Lambda$ is the set $$\mathbb{W}^s_{ \operatorname{loc},r}(0_x, \widehat{f}) \subseteq T_x M$$ of vectors $v$ with the property that $\|\widehat{f}^k(v) \| \le r$ for all $k \ge 0$ and   $\lim_{k \to \infty} \|\widehat{f}^k(v) \| = 0$.
2. The local stable manifold of $f$ at $x \in \Lambda$ is the set $$W^s_{\operatorname{loc},r}(x,f) \subseteq M$$ of points $y \in M$ such that $d(f^k(y),f^k(x)) \le r$ for all $k \ge 0$ and  $\lim_{k \to \infty} d(f^k(y),f^k(x)) = 0$.

The two notions of stable manifold are related by the exponential map: $$\exp_x \Big( \mathbb{W}^s_{ \operatorname{loc},r}(0_x, \widehat{f}) \Big) = W^s_{\operatorname{loc},r}(x,f).$$

The Stable Manifold Theorem asserts that the sets $W^s_{\operatorname{loc},r}(x,f)$ are all embedded submanifolds. We prove this by first showing that the local fibre stable manifolds $\mathbb{W}^s_{ \operatorname{loc},r}(0_x, \widehat{f})$ are embedded submanifolds of $T_x M$. This is easier than proving directly that the sets $W^s_{\operatorname{loc},r}(x,f)$ are submanifolds, since submanifolds of a vector space can be thought of as graphs. Finally, we exploit the fact that $\exp_x$ is a diffeomorphism on a small enough ball to conclude that $W^s_{\operatorname{loc},r}(x,f)$ is an embedded submanifold of $M$.