Limit of uniformly converging volume-preserving homeomorphisms

Solution 1:

This statement is not correct, the limit might not be a homeomorphism. Here is a sketch of an example in the plane: Pick a sequence of concentric circles $C_n$ of radius $1/n$ centered at $0$, let $D_n$ be the disk bounded by $C_n$, and $A_n$ the annulus between $C_n$ and $C_{n+1}$. Now pick a corresponding sequence of nested ellipses $E_n$, with $E_1 = C_1$ the unit circle, such that the area enclosed by $E_n$ is the same as the one enclosed by $C_n$, and such that the ellipses converge to a non-trivial interval $I$, e.g., $I = [-1/2, 1/2] \times \{0\}$. Let $F_n$ be the domain bounded by the ellipse $E_n$, and let $B_n$ be the topological annulus between $E_n$ and $E_{n+1}$. Then $F_n$ has the same area as $D_n$, and $B_n$ has the same area as $A_n$. Pick a sequence of orientation-preserving diffeomorphisms $\phi_n : E_n \to C_n$ with $\phi_1 = \textrm{id}$. By a classical result there exist area-preserving maps $g_n: F_n \to D_n$ (ellipse to disk) and $h_n: B_n \to A_n$ (elliptical annulus to round annulus) with boundary values given by $\phi_n$ and $\phi_{n+1}$. We also define $h_0 = \textrm{id}$ outside of the unit disk $C_1 = E_1$. Now define $f_n$ to agree with $h_k$ for $0 \le k < n$ outside of $E_n$, and with $g_n$ inside of $E_n$. Then it is easy to check that $(f_n)$ converges uniformly to an area-preserving map $f$ of the plane which maps the whole interval $I$ to the point $0$.