Hofer's length spectrum of symplectic surfaces

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2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219

Hofer's length spectrum of symplectic surfaces

Michael Khanevsky ^1,

1.	Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States

Received February 2015 Published September 2015

Abstract
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Following a question of F. Le Roux, we consider a system of invariants $l_A : H_1 (M) \to \mathbb{R}$ of a symplectic surface $M$. These invariants compute the minimal Hofer energy needed to translate a disk of area $A$ along a given homology class and can be seen as a symplectic analogue of the Riemannian length spectrum. When M has genus zero we also construct Hofer- and $C^0$-continuous quasimorphisms $Ham(M) \to H_1(M)$ that compute trajectories of periodic non-displaceable disks.

Keywords: dynamics on surfaces, Hamiltonian dynamics, length spectrum., Hofer’s metric.

Mathematics Subject Classification: Primary: 53D05; Secondary: 37J0.

Citation: Michael Khanevsky. Hofer's length spectrum of symplectic surfaces. Journal of Modern Dynamics, 2015, 9: 219-235. doi: 10.3934/jmd.2015.9.219

References:

[1]	A. Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications, 400, Springer-Verlag, 1997. doi: 10.1007/978-1-4757-6800-8.
[2]	P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math., 6 (2004), 793-802. doi: 10.1142/S0219199704001525.
[3]	D. Calegari, Word length in surface groups with characteristic generating sets, Proc. Amer. Math. Soc., 136 (2008), 2631-2637. doi: 10.1090/S0002-9939-08-09443-4.
[4]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[5]	M. Entov, L. Polterovich, P. Py and M. Khanevsky, On continuity of quasimorphisms for symplectic maps, in Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, 296, Birkhäuser, Boston, 2012. doi: 10.1007/978-0-8176-8277-4_8.
[6]	B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 2011. doi: 10.1515/9781400839049.
[7]	M. Khanevsky, Hofer's norm and disk translations in an annulus, preprint, arXiv:1111.1923.
[8]	M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory, Ph.D. thesis, Tel Aviv University, 2011.
[9]	F. Lalonde and D. McDuff, Hofer's $l^\infty$-geometry: Energy and stability of Hamiltonian flows. Part II, Invent. Math., 122 (1995), 35-69.
[10]	J.-P. Otal, Le spectre marqué des longueurs des surfaces á courbure négative, Ann. Math. (2), 131 (1990), 151-162. doi: 10.2307/1971511.
[11]	F. Le Roux, Six questions, a proposition and two pictures on Hofer distance for Hamiltonian diffeomorphisms on surfaces, in Symplectic Topology and Measure Preserving Dynamical Systems (eds. Y.-G. Oh, A. Fathi and C. Viterbo), Contemporary Mathematics, 512, Amer. Math. Soc., Providence, RI, 2010, 33-40.

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References:

[1]	A. Banyaga, The Structure of Classical Diffeomorphism Groups, Mathematics and Its Applications, 400, Springer-Verlag, 1997. doi: 10.1007/978-1-4757-6800-8.
[2]	P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math., 6 (2004), 793-802. doi: 10.1142/S0219199704001525.
[3]	D. Calegari, Word length in surface groups with characteristic generating sets, Proc. Amer. Math. Soc., 136 (2008), 2631-2637. doi: 10.1090/S0002-9939-08-09443-4.
[4]	M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not., 2003 (2003), 1635-1676. doi: 10.1155/S1073792803210011.
[5]	M. Entov, L. Polterovich, P. Py and M. Khanevsky, On continuity of quasimorphisms for symplectic maps, in Perspectives in Analysis, Geometry, and Topology, Progress in Mathematics, 296, Birkhäuser, Boston, 2012. doi: 10.1007/978-0-8176-8277-4_8.
[6]	B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 2011. doi: 10.1515/9781400839049.
[7]	M. Khanevsky, Hofer's norm and disk translations in an annulus, preprint, arXiv:1111.1923.
[8]	M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory, Ph.D. thesis, Tel Aviv University, 2011.
[9]	F. Lalonde and D. McDuff, Hofer's $l^\infty$-geometry: Energy and stability of Hamiltonian flows. Part II, Invent. Math., 122 (1995), 35-69.
[10]	J.-P. Otal, Le spectre marqué des longueurs des surfaces á courbure négative, Ann. Math. (2), 131 (1990), 151-162. doi: 10.2307/1971511.
[11]	F. Le Roux, Six questions, a proposition and two pictures on Hofer distance for Hamiltonian diffeomorphisms on surfaces, in Symplectic Topology and Measure Preserving Dynamical Systems (eds. Y.-G. Oh, A. Fathi and C. Viterbo), Contemporary Mathematics, 512, Amer. Math. Soc., Providence, RI, 2010, 33-40.

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