Stability is not open

Kai Cieliebak[1]; Urs Frauenfelder[2]; Gabriel P. Paternain[3]

  • [1] Ludwig-Maximilians-Universität Mathematisches Institut 80333 München (Germany)
  • [2] Seoul National University Department of Mathematics Research Institute of Mathematics 151-747 Seoul (South Korea)
  • [3] University of Cambridge Department of Pure Mathematics and Mathematical Statistics Cambridge CB3 0WB (UK)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 7, page 2449-2459
  • ISSN: 0373-0956

Abstract

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We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.

How to cite

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Cieliebak, Kai, Frauenfelder, Urs, and Paternain, Gabriel P.. "Stability is not open." Annales de l’institut Fourier 60.7 (2010): 2449-2459. <http://eudml.org/doc/116341>.

@article{Cieliebak2010,
abstract = {We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.},
affiliation = {Ludwig-Maximilians-Universität Mathematisches Institut 80333 München (Germany); Seoul National University Department of Mathematics Research Institute of Mathematics 151-747 Seoul (South Korea); University of Cambridge Department of Pure Mathematics and Mathematical Statistics Cambridge CB3 0WB (UK)},
author = {Cieliebak, Kai, Frauenfelder, Urs, Paternain, Gabriel P.},
journal = {Annales de l’institut Fourier},
keywords = {Stability; Hamiltonian structure; characteristic foliation; stability; Anosov Hamiltonian structure},
language = {eng},
number = {7},
pages = {2449-2459},
publisher = {Association des Annales de l’institut Fourier},
title = {Stability is not open},
url = {http://eudml.org/doc/116341},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Cieliebak, Kai
AU - Frauenfelder, Urs
AU - Paternain, Gabriel P.
TI - Stability is not open
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 7
SP - 2449
EP - 2459
AB - We give an example of a symplectic manifold with a stable hypersurface such that nearby hypersurfaces are typically unstable.
LA - eng
KW - Stability; Hamiltonian structure; characteristic foliation; stability; Anosov Hamiltonian structure
UR - http://eudml.org/doc/116341
ER -

References

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  1. D. V. Anosov, Ja. G. Sinaĭ, Certain smooth ergodic systems, Uspehi Mat. Nauk 22 (1967), 107-172 Zbl0177.42002MR224771
  2. F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799-888 Zbl1131.53312MR2026549
  3. Kai Cieliebak, Urs Adrian Frauenfelder, A Floer homology for exact contact embeddings, Pacific J. Math. 239 (2009), 251-316 Zbl1221.53112MR2461235
  4. Kai Cieliebak, Urs Adrian Frauenfelder, Gabriel P. Paternain, Symplectic topology of Mañé’s critical values, Geometry and Topology 14 (2010), 1765-1870 Zbl1239.53110MR2679582
  5. Kai Cieliebak, K. Mohnke, Compactness for punctured holomorphic curves, J. Symplectic Geom. 3 (2005), 589-654 Zbl1113.53053MR2235856
  6. Kai Cieliebak, E. Volkov, First steps in stable Hamiltonian topology, (2010) Zbl1315.53097
  7. Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000), 560-673 Zbl0989.81114MR1826267
  8. Renato Feres, Geodesic flows on manifolds of negative curvature with smooth horospheric foliations, Ergodic Theory Dynam. Systems 11 (1991), 653-686 Zbl0727.58035MR1145615
  9. Boris Hasselblatt, Horospheric foliations and relative pinching, J. Differential Geom. 39 (1994), 57-63 Zbl0795.53026MR1258914
  10. Boris Hasselblatt, Regularity of the Anosov splitting and of horospheric foliations, Ergodic Theory Dynam. Systems 14 (1994), 645-666 Zbl0821.58032MR1304137
  11. M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds, (1977), Springer-Verlag, Berlin Zbl0355.58009MR501173
  12. Helmut Hofer, Eduard Zehnder, Symplectic invariants and Hamiltonian dynamics, (1994), Birkhäuser Verlag, Basel Zbl0805.58003MR1306732
  13. Masahiko Kanai, Differential-geometric studies on dynamics of geodesic and frame flows, Japan. J. Math. (N.S.) 19 (1993), 1-30 Zbl0798.58055MR1231509
  14. Anatole Katok, Boris Hasselblatt, Introduction to the modern theory of dynamical systems, 54 (1995), Cambridge University Press, Cambridge Zbl0878.58019MR1326374
  15. William Parry, Synchronisation of canonical measures for hyperbolic attractors, Comm. Math. Phys. 106 (1986), 267-275 Zbl0618.58026MR855312
  16. Gabriel P. Paternain, Geodesic flows, 180 (1999), Birkhäuser Boston Inc., Boston, MA Zbl0930.53001MR1712465
  17. Joseph F. Plante, Anosov flows, Amer. J. Math. 94 (1972), 729-754 Zbl0257.58007MR377930
  18. Victoria Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Res. Lett. 12 (2005), 425-441 Zbl1081.37015MR2150895

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