In the last lecture we showed that if we make a $C^1$ perturbation of a map in a neighbourhood of a hyperbolic fixed point, then the new map has a unique fixed point in this neighbourhood. That is, the fixed point cannot “disappear” under perturbation. Today we upgrade this statement by showing that the new fixed point is also hyperbolic for the perturbed map.

The Local Persistence Theorem: Let $f \colon\Omega \subseteq E \to E$ be a local differentiable dynamical system. Suppose $u$ is a hyperbolic fixed point of $f$. Then any nearby map $g$ has a unique fixed point near $u$. Moreover this fixed point is hyperbolic for $g$, and the hyperbolic splitting varies continuously with $g$.

The only new statement is that the fixed point is hyperbolic, and that the hyperbolic splitting is continuous. To prove this we use a version of the Inverse Function Theorem for bi-Lipschitz maps, together with a parametric version of the Banach Fixed Point Theorem.