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Horospherically invariant measures and finitely generated Kleinian groups
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Non-autonomous curves on surfaces
On the relation between action and linking
1. | Ruhr-Universität Bochum, Universitätsstrasse 150, IB 3/79, Bochum 44801, Germany |
2. | RWTH Aachen, Jakobstrasse 2, Aachen 52064, Germany |
3. | Instituto de Matemática e Estatística, Departamento de Matemática, Universidade de São Paulo, Rua do Matão, 1010, Cidade Universitária, São Paulo, SP 05508-090, Brazil and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North, Shanghai, 200062, China |
We introduce numerical invariants of contact forms in dimension three and use asymptotic cycles to estimate them. As a consequence, we prove a version for Anosov Reeb flows of results due to Hutchings and Weiler on mean actions of periodic points. The main tool is the Action-Linking Lemma, expressing the contact area of a surface bounded by periodic orbits as the Liouville average of the asymptotic intersection number of most trajectories with the surface.
References:
[1] |
A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, preprint, arXiv: 1912.04187. |
[2] |
A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão,
Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., 211 (2018), 687-778.
doi: 10.1007/s00222-017-0755-z. |
[3] |
A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão,
Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere, Compos. Math., 154 (2018), 2643-2680.
doi: 10.1112/S0010437X18007558. |
[4] |
D. Bechara Senior, Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk, preprint, arXiv: 2003.05225. |
[5] |
G. Benedetti and J. Kang,
A local contact systolic inequality in dimension three, J. Eur. Math. Soc. (JEMS), 23 (2021), 721-764.
doi: 10.4171/jems/1022. |
[6] |
D. Cristofaro-Gardiner and M. Hutchings,
From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
|
[7] |
D. Cristofaro-Gardiner, M. Hutchings and V. G. B. Ramos,
The asymptotics of ECH capacities, Invent. Math., 199 (2015), 187-214.
doi: 10.1007/s00222-014-0510-7. |
[8] |
P. Dehornoy, Asymptotic invariants of $3$-dimensional vector fields, Winter Braids Lect. Notes, 2 (2015), 19 pp.
doi: 10.5802/wbln.8. |
[9] |
É. Ghys,
Right-handed vector fields & the Lorenz attractor, Jpn. J. Math., 4 (2009), 47-61.
doi: 10.1007/s11537-009-0854-8. |
[10] |
V. L. Ginzburg, D. Hein, U. L. Hryniewicz and L. Macarini,
Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam., 38 (2013), 55-78.
doi: 10.1007/s40306-012-0002-z. |
[11] |
U. L. Hryniewicz, A note on Schwartzman-Fried-Sullivan Theory, with an application, J. Fixed Point Theory Appl., 22 (2020), 20 pp.
doi: 10.1007/s11784-020-0757-0. |
[12] |
M. Hutchings,
Mean action and the Calabi invariant, J. Mod. Dyn., 10 (2016), 511-539.
doi: 10.3934/jmd.2016.10.511. |
[13] |
M. Hutchings, ECH capacities and the Ruelle invariant, preprint, arXiv: 1910.08260. |
[14] |
K. Irie, Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows, preprint, arXiv: 1812.01869. |
[15] |
K. Sigmund,
On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37.
doi: 10.2307/2373591. |
[16] |
M. Weiler, Mean action of periodic orbits of area-preserving annulus diffeomorphisms, J. Topol. Anal., online ready.
doi: 10.1142/S1793525320500363. |
[17] |
C. Viterbo,
Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431.
doi: 10.1090/S0894-0347-00-00328-3. |
show all references
References:
[1] |
A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, preprint, arXiv: 1912.04187. |
[2] |
A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão,
Sharp systolic inequalities for Reeb flows on the three-sphere, Invent. Math., 211 (2018), 687-778.
doi: 10.1007/s00222-017-0755-z. |
[3] |
A. Abbondandolo, B. Bramham, U. L. Hryniewicz and P. A. S. Salomão,
Systolic ratio, index of closed orbits and convexity for tight contact forms on the three-sphere, Compos. Math., 154 (2018), 2643-2680.
doi: 10.1112/S0010437X18007558. |
[4] |
D. Bechara Senior, Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk, preprint, arXiv: 2003.05225. |
[5] |
G. Benedetti and J. Kang,
A local contact systolic inequality in dimension three, J. Eur. Math. Soc. (JEMS), 23 (2021), 721-764.
doi: 10.4171/jems/1022. |
[6] |
D. Cristofaro-Gardiner and M. Hutchings,
From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.
|
[7] |
D. Cristofaro-Gardiner, M. Hutchings and V. G. B. Ramos,
The asymptotics of ECH capacities, Invent. Math., 199 (2015), 187-214.
doi: 10.1007/s00222-014-0510-7. |
[8] |
P. Dehornoy, Asymptotic invariants of $3$-dimensional vector fields, Winter Braids Lect. Notes, 2 (2015), 19 pp.
doi: 10.5802/wbln.8. |
[9] |
É. Ghys,
Right-handed vector fields & the Lorenz attractor, Jpn. J. Math., 4 (2009), 47-61.
doi: 10.1007/s11537-009-0854-8. |
[10] |
V. L. Ginzburg, D. Hein, U. L. Hryniewicz and L. Macarini,
Closed Reeb orbits on the sphere and symplectically degenerate maxima, Acta Math. Vietnam., 38 (2013), 55-78.
doi: 10.1007/s40306-012-0002-z. |
[11] |
U. L. Hryniewicz, A note on Schwartzman-Fried-Sullivan Theory, with an application, J. Fixed Point Theory Appl., 22 (2020), 20 pp.
doi: 10.1007/s11784-020-0757-0. |
[12] |
M. Hutchings,
Mean action and the Calabi invariant, J. Mod. Dyn., 10 (2016), 511-539.
doi: 10.3934/jmd.2016.10.511. |
[13] |
M. Hutchings, ECH capacities and the Ruelle invariant, preprint, arXiv: 1910.08260. |
[14] |
K. Irie, Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows, preprint, arXiv: 1812.01869. |
[15] |
K. Sigmund,
On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37.
doi: 10.2307/2373591. |
[16] |
M. Weiler, Mean action of periodic orbits of area-preserving annulus diffeomorphisms, J. Topol. Anal., online ready.
doi: 10.1142/S1793525320500363. |
[17] |
C. Viterbo,
Metric and isoperimetric problems in symplectic geometry, J. Amer. Math. Soc., 13 (2000), 411-431.
doi: 10.1090/S0894-0347-00-00328-3. |
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