American Institute of Mathematical Sciences

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

Survival of infinitely many critical points for the Rabinowitz action functional

 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.
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.

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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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