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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

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.  Google Scholar

[2]

P. Albers and U. Frauenfelder, Infinitely many leaf-wise intersection points on cotangent bundles, (2008), arXiv:0812.4426. Google Scholar

[3]

P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz Floer homology and global Hamiltonian perturbation, Journal of Modern dynamics, 4 (2010), 329-357.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

P. Albers and M. McLean, Non-displaceable contact embeddings and infinitely many leaf-wise intersections,, to appear in Journal of Symplectic Topology, ().   Google Scholar

[7]

A. Banyaga, Fixed points of symplectic maps, Invent. Math., 50 (1980), 215-229. doi: 10.1007/BF01390045.  Google Scholar

[8]

K. Cieliebak and U. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251.316.2  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[12]

V. L. Ginzburg, Coisotropic intersections, Duke Math. J., 140 (2007), 111-163. doi: 10.1215/S0012-7094-07-14014-6.  Google Scholar

[13]

B. Gürel, Leafwise coisotropic intersections, Int. Math. Res. Not., (2010), 914-931.  Google Scholar

[14]

J. Kang, Existence of leafwise intersection points in the unrestricted case, to appear in Israel Journal of Mathematics, (2009), arXiv:0910.2369 Google Scholar

[15]

J. Kang, Generalized Rabinowitz Floer homology and coisotropic intersections, (2010), arXiv:1003.1009. Google Scholar

[16]

J. Kang, Künneth formula in Rabinowitz Floer homology, (2010), arXiv:1006.0810. Google Scholar

[17]

W. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, to appear in Calc. Var. Partial Differential Equations, (2010), arXiv:1002.0162. Google Scholar

[18]

J. Moser, A fixed point theorem in symplectic geometry, Acta Math., 141 (1978), 17-34. doi: 10.1007/BF02545741.  Google Scholar

[19]

F. Ziltener, Coisotropic submanifolds, leafwise fixed points, and presymplectic embeddings, Journal of Symplectic Geom., 8 (2010), 95-118.  Google Scholar

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.  Google Scholar

[2]

P. Albers and U. Frauenfelder, Infinitely many leaf-wise intersection points on cotangent bundles, (2008), arXiv:0812.4426. Google Scholar

[3]

P. Albers and U. Frauenfelder, Spectral invariants in Rabinowitz Floer homology and global Hamiltonian perturbation, Journal of Modern dynamics, 4 (2010), 329-357.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[6]

P. Albers and M. McLean, Non-displaceable contact embeddings and infinitely many leaf-wise intersections,, to appear in Journal of Symplectic Topology, ().   Google Scholar

[7]

A. Banyaga, Fixed points of symplectic maps, Invent. Math., 50 (1980), 215-229. doi: 10.1007/BF01390045.  Google Scholar

[8]

K. Cieliebak and U. Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math., 239 (2009), 251.316.2  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[12]

V. L. Ginzburg, Coisotropic intersections, Duke Math. J., 140 (2007), 111-163. doi: 10.1215/S0012-7094-07-14014-6.  Google Scholar

[13]

B. Gürel, Leafwise coisotropic intersections, Int. Math. Res. Not., (2010), 914-931.  Google Scholar

[14]

J. Kang, Existence of leafwise intersection points in the unrestricted case, to appear in Israel Journal of Mathematics, (2009), arXiv:0910.2369 Google Scholar

[15]

J. Kang, Generalized Rabinowitz Floer homology and coisotropic intersections, (2010), arXiv:1003.1009. Google Scholar

[16]

J. Kang, Künneth formula in Rabinowitz Floer homology, (2010), arXiv:1006.0810. Google Scholar

[17]

W. Merry, On the Rabinowitz Floer homology of twisted cotangent bundles, to appear in Calc. Var. Partial Differential Equations, (2010), arXiv:1002.0162. Google Scholar

[18]

J. Moser, A fixed point theorem in symplectic geometry, Acta Math., 141 (1978), 17-34. doi: 10.1007/BF02545741.  Google Scholar

[19]

F. Ziltener, Coisotropic submanifolds, leafwise fixed points, and presymplectic embeddings, Journal of Symplectic Geom., 8 (2010), 95-118.  Google Scholar

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