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

Hofer's length spectrum of symplectic surfaces

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.
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, (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. doi: 10.1142/S0219199704001525.

[3]

D. Calegari, Word length in surface groups with characteristic generating sets,, Proc. Amer. Math. Soc., 136 (2008), 2631. 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. 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, (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, (2011). doi: 10.1515/9781400839049.

[7]

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

[8]

M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory,, Ph.D. thesis, (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.

[10]

J.-P. Otal, Le spectre marqué des longueurs des surfaces á courbure négative,, Ann. Math. (2), 131 (1990), 151. 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, (2010), 33.

show all references

References:
[1]

A. Banyaga, The Structure of Classical Diffeomorphism Groups,, Mathematics and Its Applications, (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. doi: 10.1142/S0219199704001525.

[3]

D. Calegari, Word length in surface groups with characteristic generating sets,, Proc. Amer. Math. Soc., 136 (2008), 2631. 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. 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, (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, (2011). doi: 10.1515/9781400839049.

[7]

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

[8]

M. Khanevsky, Geometric and Topological Aspects of Lagrangian Submanifolds - Intersections, Diameter and Floer Theory,, Ph.D. thesis, (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.

[10]

J.-P. Otal, Le spectre marqué des longueurs des surfaces á courbure négative,, Ann. Math. (2), 131 (1990), 151. 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, (2010), 33.

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