May  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. Google Scholar

[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. Google Scholar

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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. Google Scholar

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

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[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. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[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. Google Scholar

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[11]

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

[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. Google Scholar

[13]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[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. Google Scholar
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[37]

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

[38]

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

[39]

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

[40]

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

[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. Google Scholar

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[51]

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

[52]

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

[53]

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

[54]

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

[55]

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

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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