American Institute of Mathematical Sciences

Advanced
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

[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

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

[1]

François Ledrappier. Erratum: On Omri Sarig's work on the dynamics of surfaces. Journal of Modern Dynamics, 2015, 9: 355-355. doi: 10.3934/jmd.2015.9.355
[2]

François Ledrappier. On Omri Sarig's work on the dynamics on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 15-24. doi: 10.3934/jmd.2014.8.15
[3]

Jaeyoo Choy, Hahng-Yun Chu. On the dynamics of flows on compact metric spaces. Communications on Pure & Applied Analysis, 2010, 9 (1) : 103-108. doi: 10.3934/cpaa.2010.9.103
[4]

Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599
[5]

Dmitry Dolgopyat, Dmitry Jakobson. On small gaps in the length spectrum. Journal of Modern Dynamics, 2016, 10: 339-352. doi: 10.3934/jmd.2016.10.339
[6]

Emmanuel Schenck. Exponential gaps in the length spectrum. Journal of Modern Dynamics, 2020, 16: 207-223. doi: 10.3934/jmd.2020007
[7]

François Lalonde, Yasha Savelyev. On the injectivity radius in Hofer's geometry. Electronic Research Announcements, 2014, 21: 177-185. doi: 10.3934/era.2014.21.177
[8]

Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581
[9]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789
[10]

Gideon Simpson, Michael I. Weinstein, Philip Rosenau. On a Hamiltonian PDE arising in magma dynamics. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 903-924. doi: 10.3934/dcdsb.2008.10.903
[11]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437
[12]

Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017
[13]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[14]

Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010
[15]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[16]

Ely Kerman. Displacement energy of coisotropic submanifolds and Hofer's geometry. Journal of Modern Dynamics, 2008, 2 (3) : 471-497. doi: 10.3934/jmd.2008.2.471
[17]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406
[18]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[19]

Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077
[20]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (136)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy