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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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

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B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 2011. doi: 10.1515/9781400839049.  Google Scholar

[7]

M. Khanevsky, Hofer's norm and disk translations in an annulus,, preprint, ().   Google Scholar

[8]

M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory, Ph.D. thesis, Tel Aviv University, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[7]

M. Khanevsky, Hofer's norm and disk translations in an annulus,, preprint, ().   Google Scholar

[8]

M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory, Ph.D. thesis, Tel Aviv University, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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