2012, 6(2): 205-249. doi: 10.3934/jmd.2012.6.205

Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

1. 

Fakultät für Mathematik, TU Dortmund, Dortmund, Germany

2. 

CMLS École Polytechnique, Palaiseau, France

3. 

Mathematisches Institut der Ludwig-Maximilian-Universität, Munich, Germany

Received  January 2012 Published  August 2012

For a closed connected manifold $N$, we construct a family of functions on the Hamiltonian group $\mathcal{G}$ of the cotangent bundle $T^*N$, and a family of functions on the space of smooth functions with compact support on $T^*N$. These satisfy properties analogous to those of partial quasimorphisms and quasistates of Entov and Polterovich. The families are parametrized by the first real cohomology of $N$. In the case $N=\mathbb{T}^n$ the family of functions on $\mathcal{G}$ coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
Citation: Alexandra Monzner, Nicolas Vichery, Frol Zapolsky. Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization. Journal of Modern Dynamics, 2012, 6 (2) : 205-249. doi: 10.3934/jmd.2012.6.205
References:
[1]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product,, Geom. Topol., 14 (2010), 1569. doi: 10.2140/gt.2010.14.1569.

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().

[3]

A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique,, Comment. Math. Helv., 53 (1978), 174. doi: 10.1007/BF02566074.

[4]

P. Bernard, Symplectic aspects of Mather theory,, Duke Math. J., 136 (2007), 401.

[5]

M. Brunella, On a theorem of Sikorav,, Ens. Math. (2), 37 (1991), 83.

[6]

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

[7]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43.

[8]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds,, Compos. Math., 145 (2009), 773. doi: 10.1112/S0010437X0900400X.

[9]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.

[10]

U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds,, Israel J. Math., 159 (2007), 1. doi: 10.1007/s11856-007-0037-3.

[11]

R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles,, Internat. J. Math., 11 (2000), 1147.

[12]

S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().

[13]

S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().

[14]

R. Leclercq, Spectral invariants in Lagrangian Floer theory,, J. Mod. Dyn., 2 (2008), 249. doi: 10.3934/jmd.2008.2.249.

[15]

J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems,, Math. Z., 207 (1991), 169. doi: 10.1007/BF02571383.

[16]

D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology,, J. Korean Math. Soc., 34 (1997), 1065.

[17]

D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory,, in, 15 (1998), 107.

[18]

A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states,, J. Topol. Anal., 3 (2011), 243. doi: 10.1142/S1793525311000581.

[19]

Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle,, J. Diff. Geom., 46 (1997), 499.

[20]

Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants,, Comm. Anal. Geom., 7 (1999), 1.

[21]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds,, in, 232 (2005), 525.

[22]

G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory,, Mosc. Math. J., 3 (2003), 593.

[23]

L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms,", Lectures in Mathematics ETH Zürich, (2001).

[24]

L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps,, Math. Res. Lett., 7 (2000), 233.

[25]

P. Py, Quelques plats pour la métrique de Hofer,, J. Reine Angew. Math., 620 (2008), 185. doi: 10.1515/CRELLE.2008.053.

[26]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419.

[27]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group,, Duke Math. J., 92 (1998), 295. doi: 10.1215/S0012-7094-98-09207-9.

[28]

K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics,", Lecture Notes in Mathematics, 1844 (2004). doi: 10.1007/b97327.

[29]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms,, Geom. Topol., 14 (2010), 2383. doi: 10.2140/gt.2010.14.2383.

[30]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions,, Topology Appl., 96 (1999), 249. doi: 10.1016/S0166-8641(98)00049-2.

[31]

Viterbo, C., Symplectic topology as the geometry of generating functions,, Math. Ann. 292 (1992), 292 (1992), 685. doi: 10.1007/BF01444643.

[32]

C. Viterbo, Symplectic homogenization,, , ().

[33]

F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().

show all references

References:
[1]

A. Abbondandolo and M. Schwarz, Floer homology of cotangent bundles and the loop product,, Geom. Topol., 14 (2010), 1569. doi: 10.2140/gt.2010.14.1569.

[2]

P. Albers, A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology,, Int. Math. Res. Not. IMRN, 2008 ().

[3]

A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique,, Comment. Math. Helv., 53 (1978), 174. doi: 10.1007/BF02566074.

[4]

P. Bernard, Symplectic aspects of Mather theory,, Duke Math. J., 136 (2007), 401.

[5]

M. Brunella, On a theorem of Sikorav,, Ens. Math. (2), 37 (1991), 83.

[6]

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

[7]

M. Entov and L. Polterovich, Quasi-states and symplectic intersections,, Comment. Math. Helv., 81 (2006), 75. doi: 10.4171/CMH/43.

[8]

M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds,, Compos. Math., 145 (2009), 773. doi: 10.1112/S0010437X0900400X.

[9]

M. Entov, L. Polterovich and F. Zapolsky, Quasi-morphisms and the Poisson bracket,, Pure Appl. Math. Q., 3 (2007), 1037.

[10]

U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds,, Israel J. Math., 159 (2007), 1. doi: 10.1007/s11856-007-0037-3.

[11]

R. Iturriaga and H. Sánchez-Morgado, A minimax selector for a class of Hamiltonians on cotangent bundles,, Internat. J. Math., 11 (2000), 1147.

[12]

S. Lanzat, "Symplectic Quasi-Morphisms and Quasi-States for Noncompact Symplectic Manifolds,", Ph. D. thesis, ().

[13]

S. Lanzat, Quasi-morphisms and symplectic quasi-states for convex symplectic manifolds,, , ().

[14]

R. Leclercq, Spectral invariants in Lagrangian Floer theory,, J. Mod. Dyn., 2 (2008), 249. doi: 10.3934/jmd.2008.2.249.

[15]

J. N. Mather, Action minimizing invariant measures for positive-definite Lagrangian systems,, Math. Z., 207 (1991), 169. doi: 10.1007/BF02571383.

[16]

D. Milinković and Y.-G. Oh, Floer homology as the stable Morse homology,, J. Korean Math. Soc., 34 (1997), 1065.

[17]

D. Milinković and Y.-G. Oh, Generating functions versus action functional. Stable Morse theory versus Floer theory,, in, 15 (1998), 107.

[18]

A. Monzner and F. Zapolsky, A comparison of symplectic homogenization and Calabi quasi-states,, J. Topol. Anal., 3 (2011), 243. doi: 10.1142/S1793525311000581.

[19]

Y.-G. Oh, Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle,, J. Diff. Geom., 46 (1997), 499.

[20]

Y.-G. Oh, Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants,, Comm. Anal. Geom., 7 (1999), 1.

[21]

Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds,, in, 232 (2005), 525.

[22]

G. P. Paternain, L. Polterovich and K. F. Siburg, Boundary rigidity for Lagrangian submanifolds, non-removable intersections, and Aubry-Mather theory,, Mosc. Math. J., 3 (2003), 593.

[23]

L. Polterovich, "The Geometry of the Group of Symplectic Diffeomorphisms,", Lectures in Mathematics ETH Zürich, (2001).

[24]

L. Polterovich and K. F. Siburg, On the asymptotic geometry of area-preserving maps,, Math. Res. Lett., 7 (2000), 233.

[25]

P. Py, Quelques plats pour la métrique de Hofer,, J. Reine Angew. Math., 620 (2008), 185. doi: 10.1515/CRELLE.2008.053.

[26]

M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds,, Pacific J. Math., 193 (2000), 419. doi: 10.2140/pjm.2000.193.419.

[27]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group,, Duke Math. J., 92 (1998), 295. doi: 10.1215/S0012-7094-98-09207-9.

[28]

K. F. Siburg, "The Principle of Least Action in Geometry and Dynamics,", Lecture Notes in Mathematics, 1844 (2004). doi: 10.1007/b97327.

[29]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms,, Geom. Topol., 14 (2010), 2383. doi: 10.2140/gt.2010.14.2383.

[30]

D. Théret, A complete proof of Viterbo's uniqueness theorem on generating functions,, Topology Appl., 96 (1999), 249. doi: 10.1016/S0166-8641(98)00049-2.

[31]

Viterbo, C., Symplectic topology as the geometry of generating functions,, Math. Ann. 292 (1992), 292 (1992), 685. doi: 10.1007/BF01444643.

[32]

C. Viterbo, Symplectic homogenization,, , ().

[33]

F. Zapolsky, On the Lagrangian Hofer geometry in symplectically aspherical manifolds,, , ().

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