Asymptotic linking of volume-preserving actions of ${mathbb R}^k$

We extend V. Arnolds theory of asymptotic linking for two volume preserving flows on a domain in ${mathbb R}^3$ and $S^3$ to volume preserving actions of ${mathbb R}^k$ and ${mathbb R}^ell$ on certain domains in ${mathbb R}^n$ and also to linking of a volume preserving action of ${mathbb R}^k$ with a closed oriented singular $ell$-dimensional submanifold in ${mathbb R}^n$, where $n=k+ell+1$. We also extend the Biot-Savart formula to higher dimensions.