Suppose $\Lambda$ is a hyperbolic set for a differentiable dynamical system $f \colon M \to M$. An isolating neighbourhood for a hyperbolic set $\Lambda$ is an open set $U$ containing $\Lambda$ such that $$\Lambda = \bigcap_{k \in\mathbb{Z}}f^k(U).$$

If an isolating neighbourhood exists, we say that $\Lambda$ is an isolated hyperbolic set. Hyperbolic fixed points are always isolated, but this is not true in general. The famous homoclinic tangles of Poincaré (which we will cover in depth in a few lectures time) are examples of non-isolated hyperbolic sets.

Today we improve the persistence results from the last two lectures under the additional assumption that our original hyperbolic set is isolated.