Real analytic diffeomorphisms of the disk

Question

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?

Answer 1

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

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 $1/2$.

Answer 2

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

Answer 3

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