Quantitative conditions for right-handedness of flows
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866909461386362880 |
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| author | Florio, Anna Hryniewicz, Umberto |
| author_facet | Florio, Anna Hryniewicz, Umberto |
| contents | We give a numerical condition for right-handedness of a dynamically convex Reeb flow on the $3$-sphere. Our condition is stated in terms of an asymptotic ratio between the amount of rotation of the linearised flow and the linking number of trajectories with a periodic orbit that spans a disk-like global surface of section. As an application, we find an explicit constant $δ_* < 0.7225$ such that if a Riemannian metric on the $2$-sphere is $δ$-pinched with $δ> δ_*$, then its geodesic flow lifts to a right-handed flow on the $3$-sphere. In particular, all finite non-empty collections of periodic orbits of such a geodesic flow bind open books whose pages are global surfaces of section. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2106_12512 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Quantitative conditions for right-handedness of flows Florio, Anna Hryniewicz, Umberto Dynamical Systems Symplectic Geometry 37Exx, 53D25 We give a numerical condition for right-handedness of a dynamically convex Reeb flow on the $3$-sphere. Our condition is stated in terms of an asymptotic ratio between the amount of rotation of the linearised flow and the linking number of trajectories with a periodic orbit that spans a disk-like global surface of section. As an application, we find an explicit constant $δ_* < 0.7225$ such that if a Riemannian metric on the $2$-sphere is $δ$-pinched with $δ> δ_*$, then its geodesic flow lifts to a right-handed flow on the $3$-sphere. In particular, all finite non-empty collections of periodic orbits of such a geodesic flow bind open books whose pages are global surfaces of section. |
| title | Quantitative conditions for right-handedness of flows |
| topic | Dynamical Systems Symplectic Geometry 37Exx, 53D25 |
| url | https://arxiv.org/abs/2106.12512 |