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Is there any real analytic diffeomorphism from two dimensional disk to itself, except to the identity, such that whose restriction to the boundary is identity?

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    If your diffeomorphism restricts to the interior of the disk into itself, then I think the Schwarz lemma applies, and it must be a rotation.2011-11-30
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    Yes Leonardo! In my case they send the interior into itself! Would you please write me a reference for the Schwarz lemma? and explain more?2011-11-30
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    @Leandro: the question specifies the map is only *real* analytic, so it need not be *complex* analytic. So Schwarz isn't relevant unless you can argue the map has to be complex analytic.2011-11-30
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    The common element in Jonas's response and my own is that in the real analytic category there are functions that behave much like bump functions, so you have a fair bit of freedom to manipulate functions, at least at the $C^0$ level.2011-11-30
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    @Ryan: you're right. My bad!2011-12-01

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