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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 $V$?

Adterram
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2 Answers2

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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.

Alberto Abbondandolo
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    It took me some time to decode your answer; you should not be that cryptic. Do you know if this proof is written somewhere? – Anton Petrunin Apr 28 '12 at 22:50
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    To Anton. You are right, this is my first answer on MO and I still have to get familiar with its style. I should have added at least 2 things:

    1. The fact that every orientation preserving diffeomorphism of $\mathbb{R}^n$ is isotopic to the identity is proved in Milnor, "Topology from the differentiable viewpoint", Chapter 6, Lemma 2.

    2. The desired diffeomorphism is obtained by integrating up to time 1 the modified vector field.

     – Alberto Abbondandolo Apr 29 '12 at 08:15
  • And no, I don't know a precise reference. But the statement of Palais which Igor cites in his answer is strongly related. – Alberto Abbondandolo Apr 29 '12 at 08:21
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I believe that the result is actually Theorem 5.5 in R. S. Palais, Natural operations on differential forms, Trans. Amer. Math. Soc. vol. 92 (1959) pp. 125-141, after a bit of massaging.

Igor Rivin
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