A dynamical system $f$ on a compact manifold is said to be structurally stable if any nearby dynamical system $g$ has conjugate dynamics, i.e. there exists a homeomorphism $H_g \colon M \to M$ such that $$ H_g \circ f = g \circ H_g.$$

Structural stability gives a mechanism to produce new invariant sets. Indeed, if $ \Lambda$ is a compact invariant set of $f$, then for any nearby system $g$, the set $ H_g( \Lambda)$ is a compact invariant set for $g$.

A slightly weaker notion is weak structural stability: a dynamical system $f$ is said to be weakly structurally stable on $ \Lambda$ if for any nearby system $g$, there exists a continuous injection $H_g \colon\Lambda \to M$ such that $H_g \circ f = g \circ H_g$ on $ \Lambda$.

We will prove next lecture that hyperbolicity implies weak structural stability. This is the “missing piece” of the puzzle needed to establish persistence of hyperbolicity.



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