In this final lecture we introduce a weaker notion than structural stability, known as omega stability.

A dynamical system $f$ on a compact manifold is said to be omega stable if for any sufficiently close dynamical system $g$, the restriction $f|_{\mathsf{nw}(f)}$ is topologically conjugate to the restriction $g|_{\mathsf{nw}(g)}$.

The crowning theorem of the course is (half) of the famed Omega Stability Theorem of Smale and Palis.

The Omega Stability Theorem:

Let $f$ be a dynamical system on a compact manifold. Then $f$ is omega stable if and only if $f$ satisfies Axiom A and has no basic cycles.

We prove $\Leftarrow$ only. This direction is due to Smale (1966). The other direction is harder, and was proved by Palis in 1988, building on work of Mañé.

Thank you to everyone for attending the course. Enjoy your summer vacation, and stay safe.

Comments and questions?