Let $\Omega \subset \mathbb{ R}^n$ be an open set and $f_k \in W^{ s,p}(\Omega;\mathbb{ R}^n)$ be a sequence of homeomorphisms weakly converging to $f \in W^{ s,p}(\Omega;\mathbb{ R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is injective almost everywhere in the domain and the target. In this note we extend such results to the case $s\in(0,1)$ and $sp > n-1$. This in particular applies to $C^s$-H\"older maps.
Categories: math.AP