Mean action and the Calabi invariant

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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

Abstract
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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,, , ().
[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.
[3]	V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions,, , ().
[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.
[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.
[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.
[7]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[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.
[9]	J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1-19.
[10]	J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.
[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.
[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.
[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.
[14]	M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.
[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.
[16]	M. Hutchings, Embedded contact homology as a (symplectic) field theory,, in preparation., ().
[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.
[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.
[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.
[20]	P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511543111.
[21]	R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224.
[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.

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References:

[1]	A. Abbondandolo, B. Bramham, U. Hryniewicz and P. Salamão, Sharp systolic inequalities for Reeb flows on the three-sphere,, , ().
[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.
[3]	V. Colin, P. Ghiggini and K. Honda, Embedded contact homology and open book decompositions,, , ().
[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.
[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.
[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.
[7]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[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.
[9]	J. Franks, Area preserving homeomorphisms of open surfaces of genus zero, New York J. Math., 2 (1996), 1-19.
[10]	J.-M. Gambaudo and É. Ghys, Enlacements asymptotiques, Topology, 36 (1997), 1355-1379. doi: 10.1016/S0040-9383(97)00001-3.
[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.
[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.
[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.
[14]	M. Hutchings, Quantitative embedded contact homology, J. Diff. Geom., 88 (2011), 231-266.
[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.
[16]	M. Hutchings, Embedded contact homology as a (symplectic) field theory,, in preparation., ().
[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.
[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.
[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.
[20]	P. B. Kronheimer and T. S. Mrowka, Monopoles and Three-Manifolds, Cambridge Univ. Press, 2007. doi: 10.1017/CBO9780511543111.
[21]	R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math., 61 (2008), 1631-1684. doi: 10.1002/cpa.20224.
[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.

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