Let $f$ be a diffeomorphism on the circle $\mathbb{T}$, where $f$ is in the $C^1$ topology and has a fixed point. It is asked to prove that $f$ can be approximated by a Morse-Smale diffeomorphism by using Sard's theorem.
Approximation by a Morse-Smale diffeomorphism on the cicle
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dynamical-systems