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2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511

Mean action and the Calabi invariant

Michael Hutchings 1,

1. 

Department of Mathematics, 970 Evans Hall, University of California, Berkeley, CA 94720, United States

Received  December 2015 Revised  August 2016 Published  November 2016

Given an area-preserving diffeomorphism of the closed unit disk which is a rotation near the boundary, one can naturally define an ``action'' function on the disk which agrees with the rotation number on the boundary. The Calabi invariant of the diffeomorphism is the average of the action function over the disk. Given a periodic orbit of the diffeomorphism, its ``mean action'' is defined to be the average of the action function over the orbit. We show that if the Calabi invariant is less than the boundary rotation number, then the infimum over periodic orbits of the mean action is less than or equal to the Calabi invariant. The proof uses a new filtration on embedded contact homology determined by a transverse knot, which may be of independent interest. (An analogue of this filtration can be defined for any other version of contact homology in three dimensions that counts holomorphic curves.)
Keywords: Calabi invariant, embedded contact homology..
Mathematics Subject Classification: Primary: 37J10, 53D42; Secondary: 57R17, 57R5.
Citation: Michael Hutchings. Mean action and the Calabi invariant. Journal of Modern Dynamics, 2016, 10: 511-539. doi: 10.3934/jmd.2016.10.511
References:
[1]

A. Abbondandolo, B. Bramham, U. Hryniewicz and P. Salamão, Sharp systolic inequalities for Reeb flows on the three-sphere,, , ().   Google Scholar

[2]

E. Calabi, On the group of automorphisms of a symplectic manifold, in Problems in Analysis (Lectures at the Sympos. in honor of S. Bochner, Princeton Univ., Princeton, N.J., 1969) (ed. R. C. Gunning), Princeton Univ. Press, Princeton, 1970, 1-26.  Google Scholar

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V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions,, , ().   Google Scholar

[4]

D. Cristofaro-Gardiner, The absolute gradings on embedded contact homology and Seiberg-Witten Floer cohomology, Algebr. Geom. Topol., 13 (2013), 2239-2260. doi: 10.2140/agt.2013.13.2239.  Google Scholar

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D. Cristofaro-Gardiner, M. Hutchings and V. Ramos, The asymptotics of ECH capacities, Invent. Math., 199 (2015), 187-214. doi: 10.1007/s00222-014-0510-7.  Google Scholar

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Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, in Topological Methods in Modern Mathematics, Publish or Perish, Houston, TX, 1993, 171-193.  Google Scholar

[7]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.  Google Scholar

[8]

J. Etnyre, Legendrian and transversal knots, in Handbook of Knot Theory, Elsevier B. V., Amsterdam, 2005, 105-185. doi: 10.1016/B978-044451452-3/50004-6.  Google Scholar

[9]

J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1-19.  Google Scholar

[10]

J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.  Google Scholar

[11]

H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectizations. II. Embedding controls and algebraic invariants, Geom. and Func. Anal., 5 (1995), 270-328. doi: 10.1007/BF01895669.  Google Scholar

[12]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.  Google Scholar

[13]

H. Hofer, K. Wysock and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.  Google Scholar

[14]

M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.  Google Scholar

[15]

M. Hutchings, Lecture notes on embedded contact homology, in Contact and Symplectic Topology, Bolyai Soc. Math. Stud., 26, János Bolyai Math. Soc., Budapest, 2014, 389-484. doi: 10.1007/978-3-319-02036-5_9.  Google Scholar

[16]

M. Hutchings, Embedded contact homology as a (symplectic) field theory,, in preparation., ().   Google Scholar

[17]

M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I, J. Symplectic Geom., 5 (2007), 43-137. doi: 10.4310/JSG.2007.v5.n1.a5.  Google Scholar

[18]

M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders II, J. Symplectic Geom., 7 (2009), 29-133. doi: 10.4310/JSG.2009.v7.n1.a2.  Google Scholar

[19]

M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol., 17 (2013), 2601-2688. doi: 10.2140/gt.2013.17.2601.  Google Scholar

[20]

P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511543111.  Google Scholar

[21]

R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224.  Google Scholar

[22]

C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I, Geom. Topol., 14 (2010), 2497-2581. doi: 10.2140/gt.2010.14.2497.  Google Scholar

show all references

References:
[1]

A. Abbondandolo, B. Bramham, U. Hryniewicz and P. Salamão, Sharp systolic inequalities for Reeb flows on the three-sphere,, , ().   Google Scholar

[2]

E. Calabi, On the group of automorphisms of a symplectic manifold, in Problems in Analysis (Lectures at the Sympos. in honor of S. Bochner, Princeton Univ., Princeton, N.J., 1969) (ed. R. C. Gunning), Princeton Univ. Press, Princeton, 1970, 1-26.  Google Scholar

[3]

V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions,, , ().   Google Scholar

[4]

D. Cristofaro-Gardiner, The absolute gradings on embedded contact homology and Seiberg-Witten Floer cohomology, Algebr. Geom. Topol., 13 (2013), 2239-2260. doi: 10.2140/agt.2013.13.2239.  Google Scholar

[5]

D. Cristofaro-Gardiner, M. Hutchings and V. Ramos, The asymptotics of ECH capacities, Invent. Math., 199 (2015), 187-214. doi: 10.1007/s00222-014-0510-7.  Google Scholar

[6]

Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, in Topological Methods in Modern Mathematics, Publish or Perish, Houston, TX, 1993, 171-193.  Google Scholar

[7]

M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.  Google Scholar

[8]

J. Etnyre, Legendrian and transversal knots, in Handbook of Knot Theory, Elsevier B. V., Amsterdam, 2005, 105-185. doi: 10.1016/B978-044451452-3/50004-6.  Google Scholar

[9]

J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1-19.  Google Scholar

[10]

J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.  Google Scholar

[11]

H. Hofer, K. Wysocki and E. Zehnder, Properties of pseudo-holomorphic curves in symplectizations. II. Embedding controls and algebraic invariants, Geom. and Func. Anal., 5 (1995), 270-328. doi: 10.1007/BF01895669.  Google Scholar

[12]

H. Hofer, K. Wysocki and E. Zehnder, The dynamics on three-dimensional strictly convex energy surfaces, Ann. of Math. (2), 148 (1998), 197-289. doi: 10.2307/120994.  Google Scholar

[13]

H. Hofer, K. Wysock and E. Zehnder, Finite energy foliations of tight three-spheres and Hamiltonian dynamics, Ann. of Math. (2), 157 (2003), 125-255. doi: 10.4007/annals.2003.157.125.  Google Scholar

[14]

M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.  Google Scholar

[15]

M. Hutchings, Lecture notes on embedded contact homology, in Contact and Symplectic Topology, Bolyai Soc. Math. Stud., 26, János Bolyai Math. Soc., Budapest, 2014, 389-484. doi: 10.1007/978-3-319-02036-5_9.  Google Scholar

[16]

M. Hutchings, Embedded contact homology as a (symplectic) field theory,, in preparation., ().   Google Scholar

[17]

M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I, J. Symplectic Geom., 5 (2007), 43-137. doi: 10.4310/JSG.2007.v5.n1.a5.  Google Scholar

[18]

M. Hutchings and C. H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders II, J. Symplectic Geom., 7 (2009), 29-133. doi: 10.4310/JSG.2009.v7.n1.a2.  Google Scholar

[19]

M. Hutchings and C. H. Taubes, Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol., 17 (2013), 2601-2688. doi: 10.2140/gt.2013.17.2601.  Google Scholar

[20]

P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511543111.  Google Scholar

[21]

R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224.  Google Scholar

[22]

C. H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I, Geom. Topol., 14 (2010), 2497-2581. doi: 10.2140/gt.2010.14.2497.  Google Scholar

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