If $f$ is an orientation-preserving diffeomorphism of $\mathbb R^n$ and $K$ is a compact set in $\mathbb R^n$, can we find another diffeomorphism $\tilde f$ of $\mathbb R^n$ such that:


(1)$f=\tilde f$ on a neighborhood of $K$.
(2)There is a bounded set $V$ and $\tilde f=id$ outside $V$?



The orientation-preserving diffeomorphism of $\mathbb R^n$
Yes. You may use the fact that ***f*** is isotopic to the identity to see it as the time-1 flow of a time-dependent vector field. Then you just have to modify the vector field so that it vanishes outside from a large ball.


