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Hello.

I have problems with the following exercise:

Let \(\displaystyle f\) be a diffeomorphism of a unit ball in \(\displaystyle \mathbb{R}^n\) such that \(\displaystyle f^2 = id\), and \(\displaystyle f=id\) on a certain neighbourhood of zero. Is \(\displaystyle f=id\) ?

I know it is true for \(\displaystyle n=1\). Then we deal with \(\displaystyle f: [-1,1] \rightarrow [-1,1]\) and \(\displaystyle f\) must send endpoints to endpoints and in fact it must fix them, because it is the identity on a neighbourhood of \(\displaystyle 0\). That and the fact that continuous functions map connected sets to connected sets imply that $f=id$

However, I don't know how to prove the result for higher dimensions.

Could you help me with that?

Thank you.

I have problems with the following exercise:

Let \(\displaystyle f\) be a diffeomorphism of a unit ball in \(\displaystyle \mathbb{R}^n\) such that \(\displaystyle f^2 = id\), and \(\displaystyle f=id\) on a certain neighbourhood of zero. Is \(\displaystyle f=id\) ?

I know it is true for \(\displaystyle n=1\). Then we deal with \(\displaystyle f: [-1,1] \rightarrow [-1,1]\) and \(\displaystyle f\) must send endpoints to endpoints and in fact it must fix them, because it is the identity on a neighbourhood of \(\displaystyle 0\). That and the fact that continuous functions map connected sets to connected sets imply that $f=id$

However, I don't know how to prove the result for higher dimensions.

Could you help me with that?

Thank you.

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