American Institute of Mathematical Sciences

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  2021, 17: 319-336. doi: 10.3934/jmd.2021011

On the relation between action and linking

David Bechara Senior 1, , Umberto L. Hryniewicz 2, and  Pedro A. S. Salomão 3,

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

Received  July 31, 2020 Revised  April 09, 2021 Published  July 2021

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.

Keywords: Reeb flows, ergodic theorem, periodic orbits, action, linking number.
Mathematics Subject Classification: Primary: 37D05; Secondary: 53D10.
Citation: David Bechara Senior, Umberto L. Hryniewicz, Pedro A. S. Salomão. On the relation between action and linking. Journal of Modern Dynamics, 2021, 17: 319-336. doi: 10.3934/jmd.2021011
References:
[1]

A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, preprint, arXiv: 1912.04187. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Bechara Senior, Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk, preprint, arXiv: 2003.05225. Google Scholar

[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.  Google Scholar

[6]

D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.   Google Scholar

[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.  Google Scholar

[8]

P. Dehornoy, Asymptotic invariants of $3$-dimensional vector fields, Winter Braids Lect. Notes, 2 (2015), 19 pp. doi: 10.5802/wbln.8.  Google Scholar

[9]

É. Ghys, Right-handed vector fields & the Lorenz attractor, Jpn. J. Math., 4 (2009), 47-61.  doi: 10.1007/s11537-009-0854-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hutchings, Mean action and the Calabi invariant, J. Mod. Dyn., 10 (2016), 511-539.  doi: 10.3934/jmd.2016.10.511.  Google Scholar

[13]

M. Hutchings, ECH capacities and the Ruelle invariant, preprint, arXiv: 1910.08260. Google Scholar

[14]

K. Irie, Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows, preprint, arXiv: 1812.01869. Google Scholar

[15]

K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37.  doi: 10.2307/2373591.  Google Scholar

[16]

M. Weiler, Mean action of periodic orbits of area-preserving annulus diffeomorphisms, J. Topol. Anal., online ready. doi: 10.1142/S1793525320500363.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Abbondandolo and G. Benedetti, On the local systolic optimality of Zoll contact forms, preprint, arXiv: 1912.04187. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Bechara Senior, Asymptotic action and asymptotic winding number for area-preserving diffeomorphisms of the disk, preprint, arXiv: 2003.05225. Google Scholar

[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.  Google Scholar

[6]

D. Cristofaro-Gardiner and M. Hutchings, From one Reeb orbit to two, J. Differential Geom., 102 (2016), 25-36.   Google Scholar

[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.  Google Scholar

[8]

P. Dehornoy, Asymptotic invariants of $3$-dimensional vector fields, Winter Braids Lect. Notes, 2 (2015), 19 pp. doi: 10.5802/wbln.8.  Google Scholar

[9]

É. Ghys, Right-handed vector fields & the Lorenz attractor, Jpn. J. Math., 4 (2009), 47-61.  doi: 10.1007/s11537-009-0854-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Hutchings, Mean action and the Calabi invariant, J. Mod. Dyn., 10 (2016), 511-539.  doi: 10.3934/jmd.2016.10.511.  Google Scholar

[13]

M. Hutchings, ECH capacities and the Ruelle invariant, preprint, arXiv: 1910.08260. Google Scholar

[14]

K. Irie, Equidistributed periodic orbits of $C^\infty$-generic three-dimensional Reeb flows, preprint, arXiv: 1812.01869. Google Scholar

[15]

K. Sigmund, On the space of invariant measures for hyperbolic flows, Amer. J. Math., 94 (1972), 31-37.  doi: 10.2307/2373591.  Google Scholar

[16]

M. Weiler, Mean action of periodic orbits of area-preserving annulus diffeomorphisms, J. Topol. Anal., online ready. doi: 10.1142/S1793525320500363.  Google Scholar

[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.  Google Scholar

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