In this lecture we prove that hyperbolicity implies unique weak structural stability. More precisely:

Suppose $f \colon M \to M$ is a $C^1$ dynamical system and  $ \Lambda \subseteq M$ is a compact hyperbolic set for $f$. Then for any dynamical system $g$ which is sufficiently $C^1$ close to $f$, there is a unique continuous injection $H \colon \Lambda \to M$ which is $C^0$ close to the inclusion $ \Lambda \hookrightarrow M$ such that $H \circ f = g \circ H$ on $H$.

This concludes the section of the course on the persistence of hyperbolicity. Next week we will move onto shadowing.



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