Asymptotic linking of volume-preserving actions of ${\mathbb R}^k$
Abstract
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 ${\mathbb R}^\ell$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2020
- DOI:
- 10.48550/arXiv.2008.01823
- arXiv:
- arXiv:2008.01823
- Bibcode:
- 2020arXiv200801823L
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology;
- 57K45
- E-Print:
- 33 pages. We extend Arnol'd's asymptotic linking to actions of ${\mathbb^R}^k$ and ${\mathbb^R}^\ell$. Some typographical errors were corrected and references to Schwartzman's asymptotic cycles were added