Let $T:B\to B$ be a diffeomorphism , where $B\subset\mathbb{R}^n$ is the unit ball. I want to show there exists $x\in B$ such that $|J_T(x)|=1$ ($|J_T|$ is the absolute value of the Jacobian of $T$)
Intuitively speaking, since $T$ maps the ball to itself, I`d expect that $|J_T(x)|≡1$, Since the map does not inflate the ball, so it might only perform rotations or reflections. However my intuition here must be false. My line of reasoning was as following: $Volume(B):=\int_B1dt=\int_{T^-1(B)}|J_T(x)|dx=\int_B|J_T(x)|dx$
Where the first equality is how we defined volume, the second is by changing variables (Possible since $T$ is diffeomorphism) and the last equality because $T$ maps $B$ to itself, so $T^{-1}$ must also map $B$ to itself.
From here one can conclude that either $|J_T(x)|≡1$ or that if $|J_T(x)|\neq1$ (Say $|J_T(x)|<1$ for example) there must exist a point where $|J_T(x)|>1$ or we`d get $Volume(B)>Volume(B)$ as a contradiciton.Now simply using continuouity of $|J_T|$ along with the I.V.T gets the desired result.
From this proof, it seems like there might be such a map that isn`t a combination of rotations or reflections, and that $T$ might also "inflate" areas of the ball. This is very counter-intuitive to me, and I could not imagine such mapping. I'd be glad if someone could provide with an example, and perhaps some better intuition on the behaviour of diffeomorphisms. If such an example does not exist, I'd love to see a proof.
One possible diffeomorphism from the unit ball in $\mathbb{R}^2$ to itself, which isn't rigid in the way you expect, could be described in polar coordinates as $$(r, \theta) \mapsto \left(\frac{r + r^3}{2}, \theta \right),$$ or in rectangular coordinates as $$(x, y) \mapsto \frac{1 + x^2 + y^2}{2} (x, y).$$ (The first representation was what I came up with to make it easy to show the function is bijective, whereas the second representation makes it easier to show the differentiability of the function and its inverse at the origin.)