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

We extend V. Arnold’s 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 ${\mathbb{R}}^{\mathrm{\ell }}$ 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 $\mathrm{\ell }$-dimensional submanifold in ${\mathbb{R}}^n$, where $n=k+\mathrm{\ell }+1$. We also extend the Biot-Savart formula to higher dimensions.