# 50. The Omega Stability Theorem

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?