2019, 15: 165-207. doi: 10.3934/jmd.2019018

Mather theory and symplectic rigidity

D-MATH, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland

Received  May 11, 2018 Revised  April 02, 2019 Published  July 2019

Using methods from symplectic topology, we prove existence of invariant variational measures associated to the flow $ \phi_H $ of a Hamiltonian $ H\in C^{\infty}(M) $ on a symplectic manifold $ (M, \omega) $. These measures coincide with Mather measures (from Aubry-Mather theory) in the Tonelli case. We compare properties of the supports of these measures to classical Mather measures, and we construct an example showing that their support can be extremely unstable when $ H $ fails to be convex, even for nearly integrable $ H $. Parts of these results extend work by Viterbo [54] and Vichery [52].

Using ideas due to Entov-Polterovich [22,40], we also detect interesting invariant measures for $ \phi_H $ by studying a generalization of the symplectic shape of sublevel sets of $ H $. This approach differs from the first one in that it works also for $ (M, \omega) $ in which every compact subset can be displaced. We present applications to Hamiltonian systems on $ \mathbb R^{2n} $ and twisted cotangent bundles.

Citation: Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
References:
[1]

V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12. 

[2]

V. I. Arnol'd, Mathematical problems in classical physics, in Trends and Perspectives in Applied Mathematics, Appl. Math. Sci., 100, Springer, New York, 1994, 1–20. doi: 10.1007/978-1-4612-0859-4_1.

[3]

M. Audin and J. Lafontaine, eds., Holomorphic Curves in Symplectic Geometry, Progress in Mathematics, 117, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8508-9.

[4]

G. Benedetti and A. F. Ritter, Invariance of symplectic cohomology and twisted cotangent bundles over surfaces, arXiv: 1807.02086, 2018.

[5]

G. Benedetti, The Contact Property for Magnetic Flows on Surfaces, PhD thesis, University of Cambridge, 2015.

[6]

P. Bernard, Homoclinic orbits to invariant sets of quasi-integrable exact maps, Ergodic Theory Dynam. Systems, 20 (2000), 1583-1601.  doi: 10.1017/S0143385700000870.

[7]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420. 

[8]

P. Bernard, On the Conley decomposition of Mather sets, Rev. Mat. Iberoam., 26 (2010), 115-132.  doi: 10.4171/RMI/596.

[9]

P. Biran and K. Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel J. Math., 127 (2002), 221-244.  doi: 10.1007/BF02784532.

[10]

P. Biran and K. Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv., 76 (2001), 712-753.  doi: 10.1007/s00014-001-8326-7.

[11]

P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, arXiv: 0708.4221, 2007.

[12]

P. BiranL. Polterovich and D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J., 119 (2003), 65-118.  doi: 10.1215/S0012-7094-03-11913-4.

[13]

M. R. Bisgaard, Invariants of lagrangian cobordisms via spectral numbers, Journal of Topology and Analysis, 11 (2019), 205-231.  doi: 10.1142/S1793525319500092.

[14]

A. Bounemoura and V. Kaloshin, Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom, Mosc. Math. J., 14 (2014), 181–203,426. doi: 10.17323/1609-4514-2014-14-2-181-203.

[15]

L. BuhovskyM. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Math. (N.S.), 18 (2012), 89-157.  doi: 10.1007/s00029-011-0068-9.

[16]

Y. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.

[17]

W. F. Chen, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with nondegenerate Hessian, in Twist Mappings and Their Applications, IMA Vol. Math. Appl., 44, Springer, New York, 1992, 87–94. doi: 10.1007/978-1-4613-9257-6_5.

[18]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[19]

G. ContrerasR. IturriagaG. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.  doi: 10.1007/s000390050074.

[20]

G. Dimitroglou RizellE. Goodman and A. Ivrii, Lagrangian isotopy of tori in $S^2\times S^2$ and $\mathbb{C}P^2$, Geom. Funct. Anal., 26 (2016), 1297-1358.  doi: 10.1007/s00039-016-0388-1.

[21]

Y. Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc., 4 (1991), 513-520.  doi: 10.1090/S0894-0347-1991-1102580-2.

[22]

M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics, Nonlinearity, 30 (2017), 13-34.  doi: 10.1088/0951-7715/30/1/13.

[23]

V. L. Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimensions greater than two, in Geometry and Topology in Dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math., 246, Amer. Math. Soc., Providence, RI, 1999,113–121. doi: 10.1090/conm/246/03778.

[24]

M.-R. Herman, Existence et non Existence de Tores Invariants par des Difféomorphismes Symplectiques, in Séminaire sur les Équations aux Dérivées Partielles 1987–1988, Exp. No. XIV, 24 pp., École Polytech., Palaiseau, 1988.

[25]

H. Hofer, Lusternik-Schnirelman-theory for Lagrangian intersections, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 465-499.  doi: 10.1016/S0294-1449(16)30339-0.

[26]

T. W. Hungerford, Algebra, Reprint of the 1974 original, Graduate Texts in Mathematics, 73, Springer-Verlag, New York-Berlin, 1980.

[27]

Yu. S. Ilyashenko, A criterion of steepness for analytic functions, Uspekhi Mat. Nauk, 41 (1986), 193-194. 

[28]

A. D. Ioffe, Approximate subdifferentials and applications. I. The finite-dimensional theory, Trans. Amer. Math. Soc., 281 (1984), 389-416.  doi: 10.2307/1999541.

[29]

A. Jourani, Limit superior of subdifferentials of uniformly convergent functions, Positivity, 3 (1999), 33-47.  doi: 10.1023/A:1009740914637.

[30]

N. Kryloff and N. Bogoliouboff, La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math. (2), 38 (1937), 65–113. doi: 10.2307/1968511.

[31]

R. Leclercq and F. Zapolsky, Spectral invariants for monotone lagrangians, Journal of Topology and Analysis, 10 (2018), 627-700.  doi: 10.1142/S1793525318500267.

[32]

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

[33] D. McDuff and D. Salamon, Introduction to Symplectic Topology, third edition, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017.  doi: 10.1093/oso/9780198794899.001.0001.
[34]

A. MonznerN. Vichery and F. Zapolsky, Partial quasimorphisms and quasistates on cotangent bundles and symplectic homogenization, J. Mod. Dyn., 6 (2012), 205-249.  doi: 10.3934/jmd.2012.6.205.

[35]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.  doi: 10.1090/S0002-9947-1965-0182927-5.

[36]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–66,287.

[37]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Ⅱ, Trudy Sem. Petrovsk., 1979, 5–50.

[38]

Nikolaki, Answer to MathOverflow question no. 286830, https://mathoverflow.net/questions/286830, accessed 12/29/2017.

[39]

L. Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems, 13 (1993), 357-367.  doi: 10.1017/S0143385700007410.

[40]

L. Polterovich, Symplectic intersections and invariant measures, Ann. Math. Qué., 38 (2014), 81-93.  doi: 10.1007/s40316-014-0014-2.

[41]

A. F. Ritter, Floer theory for negative line bundles via Gromov-Witten invariants, Adv. Math., 262 (2014), 1035-1106.  doi: 10.1016/j.aim.2014.06.009.

[42]

F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv., 81 (2006), 105-121.  doi: 10.4171/CMH/45.

[43]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, expanded edition, 2014.

[44]

S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270–284. doi: 10.2307/1969999.

[45]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319.  doi: 10.1215/S0012-7094-98-09207-9.

[46]

J.-C. Sikorav, Quelques propriétés des plongements lagrangiens, Mém. Soc. Math. France (N.S.), 46 (1991), 151-167. 

[47]

J.-C. Sikorav, Rigidité symplectique dans le cotangent de Tn, Duke Math. J., 59 (1989), 759-763.  doi: 10.1215/S0012-7094-89-05935-8.

[48]

J. P. Solomon, The Calabi homomorphism, Lagrangian paths and special Lagrangians, Math. Ann., 357 (2013), 1389-1424.  doi: 10.1007/s00208-013-0946-x.

[49]

A. Sorrentino, Action-minimizing Methods in Hamiltonian Dynamics, An introduction to Aubry-Mather theory, Mathematical Notes, 50, Princeton University Press, Princeton, NJ, 2015. doi: 10.1515/9781400866618.

[50]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403.  doi: 10.2140/gt.2010.14.2383.

[51]

M. Usher, Spectral numbers in Floer theories, Compos. Math., 144 (2008), 1581-1592.  doi: 10.1112/S0010437X08003564.

[52]

N. Vichery, Spectral invariants towards a non-convex Aubry-Mather theory, arXiv: 1403.2058, 2014.

[53]

C. Viterbo, Symplectic homogenization, arXiv: 0801.0206, 2008.

[54]

C. Viterbo, Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus, arXiv: 1807.09461, 2018.

[55]

F. Zapolsky, The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory, arXiv: 1507.02253, 2015.

show all references

References:
[1]

V. I. Arnol'd, Instability of dynamical systems with many degrees of freedom, Dokl. Akad. Nauk SSSR, 156 (1964), 9-12. 

[2]

V. I. Arnol'd, Mathematical problems in classical physics, in Trends and Perspectives in Applied Mathematics, Appl. Math. Sci., 100, Springer, New York, 1994, 1–20. doi: 10.1007/978-1-4612-0859-4_1.

[3]

M. Audin and J. Lafontaine, eds., Holomorphic Curves in Symplectic Geometry, Progress in Mathematics, 117, Birkhäuser Verlag, Basel, 1994. doi: 10.1007/978-3-0348-8508-9.

[4]

G. Benedetti and A. F. Ritter, Invariance of symplectic cohomology and twisted cotangent bundles over surfaces, arXiv: 1807.02086, 2018.

[5]

G. Benedetti, The Contact Property for Magnetic Flows on Surfaces, PhD thesis, University of Cambridge, 2015.

[6]

P. Bernard, Homoclinic orbits to invariant sets of quasi-integrable exact maps, Ergodic Theory Dynam. Systems, 20 (2000), 1583-1601.  doi: 10.1017/S0143385700000870.

[7]

P. Bernard, Symplectic aspects of Mather theory, Duke Math. J., 136 (2007), 401-420. 

[8]

P. Bernard, On the Conley decomposition of Mather sets, Rev. Mat. Iberoam., 26 (2010), 115-132.  doi: 10.4171/RMI/596.

[9]

P. Biran and K. Cieliebak, Lagrangian embeddings into subcritical Stein manifolds, Israel J. Math., 127 (2002), 221-244.  doi: 10.1007/BF02784532.

[10]

P. Biran and K. Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv., 76 (2001), 712-753.  doi: 10.1007/s00014-001-8326-7.

[11]

P. Biran and O. Cornea, Quantum structures for Lagrangian submanifolds, arXiv: 0708.4221, 2007.

[12]

P. BiranL. Polterovich and D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J., 119 (2003), 65-118.  doi: 10.1215/S0012-7094-03-11913-4.

[13]

M. R. Bisgaard, Invariants of lagrangian cobordisms via spectral numbers, Journal of Topology and Analysis, 11 (2019), 205-231.  doi: 10.1142/S1793525319500092.

[14]

A. Bounemoura and V. Kaloshin, Generic fast diffusion for a class of non-convex Hamiltonians with two degrees of freedom, Mosc. Math. J., 14 (2014), 181–203,426. doi: 10.17323/1609-4514-2014-14-2-181-203.

[15]

L. BuhovskyM. Entov and L. Polterovich, Poisson brackets and symplectic invariants, Selecta Math. (N.S.), 18 (2012), 89-157.  doi: 10.1007/s00029-011-0068-9.

[16]

Y. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J., 95 (1998), 213-226.  doi: 10.1215/S0012-7094-98-09506-0.

[17]

W. F. Chen, Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with nondegenerate Hessian, in Twist Mappings and Their Applications, IMA Vol. Math. Appl., 44, Springer, New York, 1992, 87–94. doi: 10.1007/978-1-4613-9257-6_5.

[18]

F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983.

[19]

G. ContrerasR. IturriagaG. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), 788-809.  doi: 10.1007/s000390050074.

[20]

G. Dimitroglou RizellE. Goodman and A. Ivrii, Lagrangian isotopy of tori in $S^2\times S^2$ and $\mathbb{C}P^2$, Geom. Funct. Anal., 26 (2016), 1297-1358.  doi: 10.1007/s00039-016-0388-1.

[21]

Y. Eliashberg, New invariants of open symplectic and contact manifolds, J. Amer. Math. Soc., 4 (1991), 513-520.  doi: 10.1090/S0894-0347-1991-1102580-2.

[22]

M. Entov and L. Polterovich, Lagrangian tetragons and instabilities in Hamiltonian dynamics, Nonlinearity, 30 (2017), 13-34.  doi: 10.1088/0951-7715/30/1/13.

[23]

V. L. Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimensions greater than two, in Geometry and Topology in Dynamics (Winston-Salem, NC, 1998/San Antonio, TX, 1999), Contemp. Math., 246, Amer. Math. Soc., Providence, RI, 1999,113–121. doi: 10.1090/conm/246/03778.

[24]

M.-R. Herman, Existence et non Existence de Tores Invariants par des Difféomorphismes Symplectiques, in Séminaire sur les Équations aux Dérivées Partielles 1987–1988, Exp. No. XIV, 24 pp., École Polytech., Palaiseau, 1988.

[25]

H. Hofer, Lusternik-Schnirelman-theory for Lagrangian intersections, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 465-499.  doi: 10.1016/S0294-1449(16)30339-0.

[26]

T. W. Hungerford, Algebra, Reprint of the 1974 original, Graduate Texts in Mathematics, 73, Springer-Verlag, New York-Berlin, 1980.

[27]

Yu. S. Ilyashenko, A criterion of steepness for analytic functions, Uspekhi Mat. Nauk, 41 (1986), 193-194. 

[28]

A. D. Ioffe, Approximate subdifferentials and applications. I. The finite-dimensional theory, Trans. Amer. Math. Soc., 281 (1984), 389-416.  doi: 10.2307/1999541.

[29]

A. Jourani, Limit superior of subdifferentials of uniformly convergent functions, Positivity, 3 (1999), 33-47.  doi: 10.1023/A:1009740914637.

[30]

N. Kryloff and N. Bogoliouboff, La théorie générale de la mesure dans son application à l'étude des systèmes dynamiques de la mécanique non linéaire, Ann. of Math. (2), 38 (1937), 65–113. doi: 10.2307/1968511.

[31]

R. Leclercq and F. Zapolsky, Spectral invariants for monotone lagrangians, Journal of Topology and Analysis, 10 (2018), 627-700.  doi: 10.1142/S1793525318500267.

[32]

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

[33] D. McDuff and D. Salamon, Introduction to Symplectic Topology, third edition, Oxford Graduate Texts in Mathematics, Oxford University Press, Oxford, 2017.  doi: 10.1093/oso/9780198794899.001.0001.
[34]

A. MonznerN. Vichery and F. Zapolsky, Partial quasimorphisms and quasistates on cotangent bundles and symplectic homogenization, J. Mod. Dyn., 6 (2012), 205-249.  doi: 10.3934/jmd.2012.6.205.

[35]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294.  doi: 10.1090/S0002-9947-1965-0182927-5.

[36]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, Uspehi Mat. Nauk, 32 (1977), 5–66,287.

[37]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Ⅱ, Trudy Sem. Petrovsk., 1979, 5–50.

[38]

Nikolaki, Answer to MathOverflow question no. 286830, https://mathoverflow.net/questions/286830, accessed 12/29/2017.

[39]

L. Polterovich, Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems, 13 (1993), 357-367.  doi: 10.1017/S0143385700007410.

[40]

L. Polterovich, Symplectic intersections and invariant measures, Ann. Math. Qué., 38 (2014), 81-93.  doi: 10.1007/s40316-014-0014-2.

[41]

A. F. Ritter, Floer theory for negative line bundles via Gromov-Witten invariants, Adv. Math., 262 (2014), 1035-1106.  doi: 10.1016/j.aim.2014.06.009.

[42]

F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv., 81 (2006), 105-121.  doi: 10.4171/CMH/45.

[43]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 151, Cambridge University Press, Cambridge, expanded edition, 2014.

[44]

S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270–284. doi: 10.2307/1969999.

[45]

K. F. Siburg, Action-minimizing measures and the geometry of the Hamiltonian diffeomorphism group, Duke Math. J., 92 (1998), 295-319.  doi: 10.1215/S0012-7094-98-09207-9.

[46]

J.-C. Sikorav, Quelques propriétés des plongements lagrangiens, Mém. Soc. Math. France (N.S.), 46 (1991), 151-167. 

[47]

J.-C. Sikorav, Rigidité symplectique dans le cotangent de Tn, Duke Math. J., 59 (1989), 759-763.  doi: 10.1215/S0012-7094-89-05935-8.

[48]

J. P. Solomon, The Calabi homomorphism, Lagrangian paths and special Lagrangians, Math. Ann., 357 (2013), 1389-1424.  doi: 10.1007/s00208-013-0946-x.

[49]

A. Sorrentino, Action-minimizing Methods in Hamiltonian Dynamics, An introduction to Aubry-Mather theory, Mathematical Notes, 50, Princeton University Press, Princeton, NJ, 2015. doi: 10.1515/9781400866618.

[50]

A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403.  doi: 10.2140/gt.2010.14.2383.

[51]

M. Usher, Spectral numbers in Floer theories, Compos. Math., 144 (2008), 1581-1592.  doi: 10.1112/S0010437X08003564.

[52]

N. Vichery, Spectral invariants towards a non-convex Aubry-Mather theory, arXiv: 1403.2058, 2014.

[53]

C. Viterbo, Symplectic homogenization, arXiv: 0801.0206, 2008.

[54]

C. Viterbo, Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus, arXiv: 1807.09461, 2018.

[55]

F. Zapolsky, The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory, arXiv: 1507.02253, 2015.

Figure 1.  Graph of $ \varphi $ indicated in red
Figure 2.  The projection of $ {\rm Supp}(\mathfrak{M}_{H_0:T(0, 0)}) $ to $ \mathbb R^2 $ is contained in the red spot in the center. For all $ \epsilon>0 $, the projection of $ {\rm Supp}(\mathfrak{M}_{H_{\epsilon}:T(0, 0)}) $ to $ \mathbb R^2 $ is contained in the blue regions
Figure 3.  On the left $ {\rm Graph}(h) $ is indicated. On the right the sublevel set $ \Sigma_{3/2} $ is indicated. The red dashed line indicates the level set $\{H = 0\}$ which consists of fixed points for $ \phi_H $. The arrows $ a $ and $ b $ indicate the direction of $ \phi_H $-flowlines
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Kei Irie. Dense existence of periodic Reeb orbits and ECH spectral invariants. Journal of Modern Dynamics, 2015, 9: 357-363. doi: 10.3934/jmd.2015.9.357

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