If $f$ is a diffeomorphism is it true that $NW(f|_{NW(f)})=NW(f)$?

Question

If $f$ is a diffeomorphism is it true that $NW(f|_{NW(f)})=NW(f)$ where $NW(f)$ is the nonwandering set of $f$?

Answer 1

In general, it is not true that $NW(f|_{NW(f)})=NW(f)$. For example, consider the time-one map $f$ of the flow of the equation in polar coordinates $\frac{dr}{dt}=r(1-r),\frac{d\theta}{dt}=\sin{\theta}^{2}+1-r^2.$ Then it's easy to check that $NW(f)$ is the union of the origin and the unit circle while $NW(f|_{NW(f)})$ consists of only three points.