Survival of infinitely many critical points for the Rabinowitz action functional

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October 2010, 4(4): 733-739. doi: 10.3934/jmd.2010.4.733

Survival of infinitely many critical points for the Rabinowitz action functional

Jungsoo Kang ^1,

1.	Department of Mathematical Sciences, Seoul National University, Kwanakgu Shinrim, San56-1 Seoul, South Korea

Received June 2010 Revised November 2010 Published January 2011

Abstract
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In this paper, we show that if Rabinowitz Floer homology has infinite dimension, there exist infinitely many critical points of a Rabinowitz action functional even though it could be non-Morse. This result is proved by examining filtered Rabinowitz Floer homology.

Keywords: leafwise intersections., Rabinowitz Floer homology.

Mathematics Subject Classification: 53D40, 37J10, 58J0.

Citation: Jungsoo Kang. Survival of infinitely many critical points for the Rabinowitz action functional. Journal of Modern Dynamics, 2010, 4 (4) : 733-739. doi: 10.3934/jmd.2010.4.733

References:

[1]	P. Albers and U. Frauenfelder, Leaf-wise intersections and Rabinowitz Floer homology, Journal of Topology and Analysis, 2 (2010), 77-98. doi: 10.1142/S1793525310000276.
[2]	P. Albers and U. Frauenfelder, Infinitely many leaf-wise intersection points on cotangent bundles, (2008), arXiv:0812.4426.
[3]	P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz Floer homology and global Hamiltonian perturbation, Journal of Modern dynamics, 4 (2010), 329-357.
[4]	P. Albers and U. Frauenfelder, A remark on a theorem by Ekeland-Hofer, (2010), to appear in Israel Journal of Mathematics, arXiv:1001.3386.
[5]	P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Mathematical Proceedings of the Cambridge Philosophical Society, 149 (2010), 539-551. doi: 10.1017/S0305004110000435.
[6]	P. Albers and M. McLean, Non-displaceable contact embeddings and infinitely many leaf-wise intersections,, to appear in Journal of Symplectic Topology, ().
[7]	A. Banyaga, Fixed points of symplectic maps, Invent. Math., 50 (1980), 215-229. doi: 10.1007/BF01390045.
[8]	K. Cieliebak and U. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251.316.2
[9]	K. Cieliebak and U. Frauenfelder, A. Oancea, Rabinowitz Floer homology and symplectic homology, to appear in Annales Scientifiques de LÉNS, (2009), arXiv:0903.0768.
[10]	D. L. Dragnev, Symplectic rigidity, symplectic fixed points and global perturbations of Hamiltonian systems, Comm. Pure Appl. Math., 61 (2008), 346-370. doi: 10.1002/cpa.20203.
[11]	I. Ekeland and H. Hofer, Two symplectic fixed-point theorems with applications to Hamiltonian dynamics, J. Math. Pures Appl., 68 (1989), 467-489 (1990).
[12]	V. L. Ginzburg, Coisotropic intersections, Duke Math. J., 140 (2007), 111-163. doi: 10.1215/S0012-7094-07-14014-6.
[13]	B. Gürel, Leafwise coisotropic intersections, Int. Math. Res. Not., (2010), 914-931.
[14]	J. Kang, Existence of leafwise intersection points in the unrestricted case, to appear in Israel Journal of Mathematics, (2009), arXiv:0910.2369
[15]	J. Kang, Generalized Rabinowitz Floer homology and coisotropic intersections, (2010), arXiv:1003.1009.
[16]	J. Kang, Künneth formula in Rabinowitz Floer homology, (2010), arXiv:1006.0810.
[17]	W. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, to appear in Calc. Var. Partial Differential Equations, (2010), arXiv:1002.0162.
[18]	J. Moser, A fixed point theorem in symplectic geometry, Acta Math., 141 (1978), 17-34. doi: 10.1007/BF02545741.
[19]	F. Ziltener, Coisotropic submanifolds, leafwise fixed points, and presymplectic embeddings, Journal of Symplectic Geom., 8 (2010), 95-118.

show all references

References:

[1]	P. Albers and U. Frauenfelder, Leaf-wise intersections and Rabinowitz Floer homology, Journal of Topology and Analysis, 2 (2010), 77-98. doi: 10.1142/S1793525310000276.
[2]	P. Albers and U. Frauenfelder, Infinitely many leaf-wise intersection points on cotangent bundles, (2008), arXiv:0812.4426.
[3]	P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz Floer homology and global Hamiltonian perturbation, Journal of Modern dynamics, 4 (2010), 329-357.
[4]	P. Albers and U. Frauenfelder, A remark on a theorem by Ekeland-Hofer, (2010), to appear in Israel Journal of Mathematics, arXiv:1001.3386.
[5]	P. Albers and A. Momin, Cup-length estimates for leaf-wise intersections, Mathematical Proceedings of the Cambridge Philosophical Society, 149 (2010), 539-551. doi: 10.1017/S0305004110000435.
[6]	P. Albers and M. McLean, Non-displaceable contact embeddings and infinitely many leaf-wise intersections,, to appear in Journal of Symplectic Topology, ().
[7]	A. Banyaga, Fixed points of symplectic maps, Invent. Math., 50 (1980), 215-229. doi: 10.1007/BF01390045.
[8]	K. Cieliebak and U. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251.316.2
[9]	K. Cieliebak and U. Frauenfelder, A. Oancea, Rabinowitz Floer homology and symplectic homology, to appear in Annales Scientifiques de LÉNS, (2009), arXiv:0903.0768.
[10]	D. L. Dragnev, Symplectic rigidity, symplectic fixed points and global perturbations of Hamiltonian systems, Comm. Pure Appl. Math., 61 (2008), 346-370. doi: 10.1002/cpa.20203.
[11]	I. Ekeland and H. Hofer, Two symplectic fixed-point theorems with applications to Hamiltonian dynamics, J. Math. Pures Appl., 68 (1989), 467-489 (1990).
[12]	V. L. Ginzburg, Coisotropic intersections, Duke Math. J., 140 (2007), 111-163. doi: 10.1215/S0012-7094-07-14014-6.
[13]	B. Gürel, Leafwise coisotropic intersections, Int. Math. Res. Not., (2010), 914-931.
[14]	J. Kang, Existence of leafwise intersection points in the unrestricted case, to appear in Israel Journal of Mathematics, (2009), arXiv:0910.2369
[15]	J. Kang, Generalized Rabinowitz Floer homology and coisotropic intersections, (2010), arXiv:1003.1009.
[16]	J. Kang, Künneth formula in Rabinowitz Floer homology, (2010), arXiv:1006.0810.
[17]	W. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, to appear in Calc. Var. Partial Differential Equations, (2010), arXiv:1002.0162.
[18]	J. Moser, A fixed point theorem in symplectic geometry, Acta Math., 141 (1978), 17-34. doi: 10.1007/BF02545741.
[19]	F. Ziltener, Coisotropic submanifolds, leafwise fixed points, and presymplectic embeddings, Journal of Symplectic Geom., 8 (2010), 95-118.

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