Part of the NATO Science Series II: Mathematics, Physics and Chemistry book series (NAII, volume 217)


After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the assumption that rest points have finite Morse index. Then we extend these ideas to rest points with infinite Morse index and co-index, by using a suitable subbundle of the tangent bundle of M as a comparison object.


Unstable Manifold Stable Manifold Morse Index Morse Theory Morse Function 
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  1. Abbondandolo, A. (1997) A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces, Topol. Methods Nonlinear Anal. 9, 325–382.zbMATHMathSciNetGoogle Scholar
  2. Abbondandolo, A. (2000) Morse theory for asymptotically linear Hamiltonian systems, Nonlinear Anal. 39, 997–1049.CrossRefzbMATHMathSciNetGoogle Scholar
  3. Abbondandolo, A. (2001) Morse Theory for Hamiltonian Systems, Vol. 425 of Chapman Hall CRC Res. Notes Math., Boca Raton, FL, Chapman & Hall/CRC.Google Scholar
  4. Abbondandolo, A. and Majer, P. (2001) Morse homology on Hilbert spaces, Comm. Pure Appl. Math. 54, 689–760.CrossRefMathSciNetzbMATHGoogle Scholar
  5. Abbondandolo, A. and Majer, P. (2003)a Infinite-dimensional Grassmannians, arXiv:math. AT/0307192.Google Scholar
  6. Abbondandolo, A. and Majer, P. (2003)b A Morse complex for infinite-dimensional manifolds. I, Adv. in Math., to appear arXiv:math. DS/0309020.Google Scholar
  7. Abbondandolo, A. and Majer, P. (2003)c Ordinary differential operators on Hilbert spaces and Fredholm pairs, Math. Z. 243, 525–562.CrossRefMathSciNetzbMATHGoogle Scholar
  8. Abbondandolo, A. and Majer, P. (2004) When the Morse index is infinite, Int. Math. Res. Not. 71, 3839–3854.MathSciNetGoogle Scholar
  9. Abraham, R. and Robbin, J. (1967) Transversal Mappings and Flows, NewYork, W. A. Benjamin, Inc.zbMATHGoogle Scholar
  10. Angenent, S. and van der Vorst, R. (1999) A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology, Math. Z. 231, 203–248.MathSciNetzbMATHGoogle Scholar
  11. Arbarello, E. (2002) Sketches of KdV, In A. Bertram, J. A. Carlson, and H. Kley (eds.), Symposium in Honor of C. H. Clemens, Vol. 312 of Contemp. Math., Salt Lake City, UT, 2000, p. 9–69, Providence, RI, Amer. Math. Soc.Google Scholar
  12. Banyaga, A. and Hurtubise, D. (2004) Lectures on Morse Homology, Vol. 29 of Kluwer Texts Math. Sci., Dordrecht, Springer.Google Scholar
  13. Benci, V. and Rabinowitz, P. H. (1979) Critical point theorems for indefinite functionals, Invent. Math. 52, 241–273.CrossRefMathSciNetzbMATHGoogle Scholar
  14. Bott, R. (1959) The stable homotopy of the classical groups, Ann. of Math. (2) 70, 313–317.zbMATHMathSciNetGoogle Scholar
  15. Bott, R. (1982) Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N. S.) 7, 331–358.zbMATHMathSciNetCrossRefGoogle Scholar
  16. Bott, R. (1988) Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68, 99–114.zbMATHMathSciNetGoogle Scholar
  17. Chang, K. C. (1981) Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34, 693–712.zbMATHMathSciNetGoogle Scholar
  18. Chang, K. C. (1993) Infinite-dimensional Morse theory and multiple solution problems, Vol. 6 of Progr. Nonlinear Differential Equations Appl., Boston, MA, Birkhäuser.Google Scholar
  19. Cohen, R. L., Jones, J. D. S., and Segal, G. B. (1995) Floer’s infinite-dimensional Morse theory and homotopy theory, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, p. 297–325, Basel, Birkhäuser.Google Scholar
  20. Conley, C. (1978) Isolated invariant sets and the Morse index, Vol. 38 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.Google Scholar
  21. Conley, C. and Zehnder, E. (1983) The Birkhoff– Lewis fixed point theorem and a conjecture of V. I. Arnol'd, Invent. Math. 73, 33–49.CrossRefMathSciNetzbMATHGoogle Scholar
  22. Conley, C. and Zehnder, E. (1984) Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37, 207–253.MathSciNetzbMATHGoogle Scholar
  23. Cornea, O. (2002)a Homotopical dynamics. II. Hopf invariants, smoothing, and the Morse complex, Ann. Sci. École Norm. Sup.(4) 35, 549–573.MathSciNetzbMATHGoogle Scholar
  24. Cornea, O. (2002)b Homotopical dynamics. IV. Hopf invariants, and Hamiltonian flows, Comm. Pure Appl. Math. 55, 1033–1088.CrossRefMathSciNetzbMATHGoogle Scholar
  25. Cornea, O. and Ranicki, A. (2003) Rigidity and glueing for Morse and Novikov complexes, J. Eur. Math. Soc. (JEMS) 5, 343–394.MathSciNetzbMATHGoogle Scholar
  26. Dold, A. (1980) Lectures on Algebraic Topology, Vol. 200 of Grundlehren Math. Wiss., Berlin, Springer, 2 edition.Google Scholar
  27. Dugundji, J. (1978) Topology, Allyn and Bacon Series in Advanced Mathematics, Boston, MA, Allyn and Bacon Inc., reprinting of the 1966 original.Google Scholar
  28. Eells, J. and Elworthy, K. D. (1970) Open embeddings of certain Banach manifolds, Ann. of Math. (2) 91, 465–485.MathSciNetGoogle Scholar
  29. Floer, A. (1988)a Morse theory for Lagrangian intersections, J. Differential Geom. 28, 513–547.MathSciNetzbMATHGoogle Scholar
  30. Floer, A. (1988)b A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41, 393–407.MathSciNetzbMATHGoogle Scholar
  31. Floer, A. (1988)c The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775–813.MathSciNetzbMATHGoogle Scholar
  32. Floer, A. (1989) Witten’s complex and infinite-dimensional Morse theory, J. Diifferential Geom. 30, 207–221.zbMATHMathSciNetGoogle Scholar
  33. Franks, J. M. (1979) Morse–Smale flows and homotopy theory, Topology 18, 199–215.CrossRefzbMATHMathSciNetGoogle Scholar
  34. Franks, J. M. (1980) Homology and Dynamical Systems, Vol. 49 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.Google Scholar
  35. Geba, K., Izydorek, M., and Pruszko, A. (1999) The Conley index in Hilbert spaces and its applications, Studia Math. 134, 217–233.MathSciNetzbMATHGoogle Scholar
  36. Guest, M. A. (1997) Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38 ofLondon Math. Soc. Stud. Texts, Cambridge, Cambridge Univ. Press.Google Scholar
  37. Harvey, F. R. and Lawson, Jr., H. B. (2001) Finite volume flows and Morse theory, Ann. of Math. (2) 153, 1–25.MathSciNetzbMATHGoogle Scholar
  38. Izydorek, M. (2001) A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Differential Equations 170, 22–50.CrossRefzbMATHMathSciNetGoogle Scholar
  39. Kato, T. (1980) Perturbation Theory for Linear Operators, Berlin, Springer, 2 edition.Google Scholar
  40. Klingenberg, W. (1978) Lectures on Closed Geodesics, Vol. 230 of Grundlehren Math. Wiss., Berlin, Springer.Google Scholar
  41. Klingenberg, W. (1982) Riemannian Geometry, Vol. 1 of de Gruyter Stud. Math., Berlin, de Gruyter.Google Scholar
  42. Kryszewski, W. and Szulkin, A. (1997) An infinite-dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349, 3181–3234.CrossRefMathSciNetzbMATHGoogle Scholar
  43. Kuiper, N. H. (1965) The homotopy type of the unitary group of Hilbert space, Topology 3, 19–30.CrossRefzbMATHMathSciNetGoogle Scholar
  44. Lang, S. (1999) Fundamentals of Differential Geometry, Vol. 191 of Grad. Texts in Math., New York, Springer.Google Scholar
  45. Lasry, J.-M. and Lions, P.-L. (1986) A remark on regularization in Hilbert spaces, Israel J. Math. 55, 257–266.MathSciNetzbMATHGoogle Scholar
  46. Mawhin, J. and Willem, M. (1989) Critical Point Theory and Hamiltonian Systems, Vol. 74 of Appl. Math. Sci., New York, Springer.Google Scholar
  47. Milnor, J. (1963) Morse Theory, Vol. 51 of Ann. of Math. Stud., Princeton, NJ, Princeton Univ. Press.Google Scholar
  48. Milnor, J. W. (1965) Topology from the Differentiable Viewpoint, Charlottesville, VA, The University Press of Virginia.zbMATHGoogle Scholar
  49. Morse, M. (1925) Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27, 345–396.zbMATHMathSciNetGoogle Scholar
  50. Morse, M. (1934) The Calculus of Variations in the Large, Vol. 18 of Amer. Math. Soc. Colloq. Publ, Providence, RI, Amer. Math. Soc.Google Scholar
  51. Morse, M. (1947) Introduction to Analysis in the Large, Princeton, NJ, Princeton Univ. Press.Google Scholar
  52. Nemirovskiĭ A. S. and Semenov, S. M. (1973) The polynomial approximation of functions in Hilbert spaces, Mat. Sb. (N. S.) 21 (92), 255–277, Russian.Google Scholar
  53. Palais, R. S. (1963) Morse theory on Hilbert manifolds, Topology, 2 299–340.CrossRefzbMATHMathSciNetGoogle Scholar
  54. Palais, R. S. (1965) On the homotopy type of certain groups of operators, Topology 3, 271–279.CrossRefzbMATHMathSciNetGoogle Scholar
  55. Palis, J. (1968) On Morse–Smale dynamical systems, Topology 8, 385–405.MathSciNetGoogle Scholar
  56. Palis, Jr., J. and de Melo, W. (1982) Geometric Theory of Dynamical Systems, New York, Springer.zbMATHGoogle Scholar
  57. Poźniak, M. (1991) The Morse complex, Novikov cohomology and Fredholm theory, preprint, University of Warwick.Google Scholar
  58. Pressley, A. and Segal, G. (1986) Loop Groups, Oxford Math. Monogr., Oxford, Oxford Univ. Press.Google Scholar
  59. Quillen, D. (1985) Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl. 19, 31–34.CrossRefzbMATHMathSciNetGoogle Scholar
  60. Rabinowitz, P. H. (1986) Minimax Methods in Critical Point Theory with Applications to Dif-ferential Equations, Vol. 65 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.Google Scholar
  61. Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1–41.zbMATHMathSciNetGoogle Scholar
  62. Salamon, D. (1990) Morse theory, the Conley index and Floer homology, Bull. Amer. Math. Soc. 22, 113–140.zbMATHMathSciNetGoogle Scholar
  63. Salamon, D. (1999) Lectures on Floer homology, In Y. Eliashberg and L. Traynor (eds.), Symplectic Geometry and Topology, Vol. 7 of IAS/ Park City Math. Ser., Park City, UT, 1997, p. 143–229, Providence, RI, Amer. Math. Soc.Google Scholar
  64. Sato, M. (1981) Soliton equations as dynamical systems on an infinite-dimensional Grassman manifold, Sūrikaisekikenkyūsho Kōkyūroku 439, 30–46.zbMATHGoogle Scholar
  65. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Mah, Basel, Birkhäuser.Google Scholar
  66. Schwarz, M. (1999) Equivalence for Morse homology, In M. Barge and K. Kuperberg (eds.), Geometry and Topology in Dynamics, Vol. 246 of Contemp. Math., Winston-Salem, NC, 1998/San Antonio, TX, 1999, p. 197–216, Providence, RI, Amer. Math. Soc.Google Scholar
  67. Segal, G. and Wilson, G. (1985) Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61, 5–65.MathSciNetzbMATHCrossRefGoogle Scholar
  68. Shub, M. (1987) Global Stability of Dynamical Systems, New York, Springer.zbMATHGoogle Scholar
  69. Smale, S. (1960) Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66, 43–49.zbMATHMathSciNetGoogle Scholar
  70. Smale, S. (1961) On gradient dynamical systems, Ann. of Math.(2) 74, 199–206.zbMATHMathSciNetGoogle Scholar
  71. Smale, S. (1964)a A generalized Morse theory, Bull. Amer. Math. Soc. 70, 165–172.MathSciNetzbMATHGoogle Scholar
  72. Smale, S. (1964)b Morse theory and a nonlinear generalization of the Dirichlet problem, Ann. of Math.(2) 80, 382–396.MathSciNetGoogle Scholar
  73. Smale, S. (1965) An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, 861–866.zbMATHMathSciNetGoogle Scholar
  74. Smale, S. (1967) Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747–817.zbMATHMathSciNetCrossRefGoogle Scholar
  75. Szulkin, A. (1992) Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, 375–418.zbMATHMathSciNetGoogle Scholar
  76. Thom, R. (1949) Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris 228, 973–975.zbMATHMathSciNetGoogle Scholar
  77. Weber, J. (1993) Der Morse–Witten Komplex, Master’s thesis, TU Berlin.Google Scholar
  78. Weber, J. (2005) The Morse–Witten complex via dynamical systems, Exposition. Math., to appear.Google Scholar
  79. Witten, E. (1982) Supersymmetry and Morse theory, J. Dierential Geom. 17, 661–692.zbMATHMathSciNetGoogle Scholar

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© Springer 2006

Authors and Affiliations

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  1. 1.Università di PisaPisa

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