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Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows
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Coles, Solly Thompson and Sharp, Richard (2022) Helicity, linking and the distribution of null-homologous periodic orbits for Anosov flows. Nonlinearity, 36 (1). 21. doi:10.1088/1361-6544/ac9f9d ISSN 0951-7715.
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Official URL: https://doi.org/10.1088/1361-6544/ac9f9d
Abstract
This paper concerns connections between dynamical systems, knots and helicity of vector fields. For a divergence-free vector field on a closed three-manifold that generates an Anosov flow, we show that the helicity of the vector field may be recovered as the limit of appropriately weighted averages of linking numbers of periodic orbits, regarded as knots. This complements a classical result of Arnold and Vogel that, when the manifold is a real homology three-sphere, the helicity may be obtained as the limit of the normalised linking numbers of typical pairs of long trajectories. We also obtain results on the asymptotic distribution of weighted averages of null-homologous periodic orbits.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics | ||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
Library of Congress Subject Headings (LCSH): | Anosov flows, Combinatorial dynamics, Vector fields | ||||||
Journal or Publication Title: | Nonlinearity | ||||||
Publisher: | Institute of Physics Publishing Ltd. | ||||||
ISSN: | 0951-7715 | ||||||
Official Date: | 17 November 2022 | ||||||
Dates: |
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Volume: | 36 | ||||||
Number: | 1 | ||||||
Article Number: | 21 | ||||||
DOI: | 10.1088/1361-6544/ac9f9d | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Access rights to Published version: | Open Access (Creative Commons) | ||||||
Date of first compliant deposit: | 8 November 2022 | ||||||
Date of first compliant Open Access: | 8 November 2022 | ||||||
RIOXX Funder/Project Grant: |
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