# Real analytic diffeomorphisms of the disk

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. – student Nov 30 '11 at 23:32
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? – Mahdi Teymuri Garakani Nov 30 '11 at 23:36
@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. – Ryan Budney Nov 30 '11 at 23:37
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. – Ryan Budney Nov 30 '11 at 23:46
@Ryan: you're right. My bad! – student Dec 1 '11 at 12:24

$f(z)=ze^{2\pi|z|^2i}$ should do the trick.

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We perturbed the identity map in different directions. – Ryan Budney Nov 30 '11 at 23:45
Thanks for the example! – Mahdi Teymuri Garakani Dec 1 '11 at 0:06

Some that spring to mind are $(x,y) \to (x+ c (1 - x^2 - y^2), y)$ where $-1/2 < c < 1/2$, $c \ne 0$.

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Thanks for the example! – Mahdi Teymuri Garakani Dec 1 '11 at 0:05

$$f(x,y) = (x,y) + (-x,-y)(x^2+y^2)(1-x^2-y^2)$$

Doesn't the above map do the job? I'm using the disc in $\mathbb R^2$ given by $x^2+y^2 \leq 1$.

If you want one without a fixed point in the interior,

$$f(x,y) = (x,y) + \left(\frac{1}{10},0\right)(1-x^2-y^2)$$

The fraction $\frac{1}{10}$ just needs to be a positive number strictly smaller than $1/2$.

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Thanks! I think the example works! Do you know any example without any fixed point in the interior? – Mahdi Teymuri Garakani Nov 30 '11 at 23:49
I'll edit in one. – Ryan Budney Nov 30 '11 at 23:52
Thanks for the example! – Mahdi Teymuri Garakani Dec 1 '11 at 0:06