Abstract
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.
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References
Abbondandolo, A. (1997) A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces, Topol. Methods Nonlinear Anal. 9, 325–382.
Abbondandolo, A. (2000) Morse theory for asymptotically linear Hamiltonian systems, Nonlinear Anal. 39, 997–1049.
Abbondandolo, A. (2001) Morse Theory for Hamiltonian Systems, Vol. 425 of Chapman Hall CRC Res. Notes Math., Boca Raton, FL, Chapman & Hall/CRC.
Abbondandolo, A. and Majer, P. (2001) Morse homology on Hilbert spaces, Comm. Pure Appl. Math. 54, 689–760.
Abbondandolo, A. and Majer, P. (2003)a Infinite-dimensional Grassmannians, arXiv:math. AT/0307192.
Abbondandolo, A. and Majer, P. (2003)b A Morse complex for infinite-dimensional manifolds. I, Adv. in Math., to appear arXiv:math. DS/0309020.
Abbondandolo, A. and Majer, P. (2003)c Ordinary differential operators on Hilbert spaces and Fredholm pairs, Math. Z. 243, 525–562.
Abbondandolo, A. and Majer, P. (2004) When the Morse index is infinite, Int. Math. Res. Not. 71, 3839–3854.
Abraham, R. and Robbin, J. (1967) Transversal Mappings and Flows, NewYork, W. A. Benjamin, Inc.
Angenent, S. and van der Vorst, R. (1999) A superquadratic indefinite elliptic system and its Morse–Conley–Floer homology, Math. Z. 231, 203–248.
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.
Banyaga, A. and Hurtubise, D. (2004) Lectures on Morse Homology, Vol. 29 of Kluwer Texts Math. Sci., Dordrecht, Springer.
Benci, V. and Rabinowitz, P. H. (1979) Critical point theorems for indefinite functionals, Invent. Math. 52, 241–273.
Bott, R. (1959) The stable homotopy of the classical groups, Ann. of Math. (2) 70, 313–317.
Bott, R. (1982) Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N. S.) 7, 331–358.
Bott, R. (1988) Morse theory indomitable, Inst. Hautes Études Sci. Publ. Math. 68, 99–114.
Chang, K. C. (1981) Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34, 693–712.
Chang, K. C. (1993) Infinite-dimensional Morse theory and multiple solution problems, Vol. 6 of Progr. Nonlinear Differential Equations Appl., Boston, MA, Birkhäuser.
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.
Conley, C. (1978) Isolated invariant sets and the Morse index, Vol. 38 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.
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.
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.
Cornea, O. (2002)a Homotopical dynamics. II. Hopf invariants, smoothing, and the Morse complex, Ann. Sci. École Norm. Sup.(4) 35, 549–573.
Cornea, O. (2002)b Homotopical dynamics. IV. Hopf invariants, and Hamiltonian flows, Comm. Pure Appl. Math. 55, 1033–1088.
Cornea, O. and Ranicki, A. (2003) Rigidity and glueing for Morse and Novikov complexes, J. Eur. Math. Soc. (JEMS) 5, 343–394.
Dold, A. (1980) Lectures on Algebraic Topology, Vol. 200 of Grundlehren Math. Wiss., Berlin, Springer, 2 edition.
Dugundji, J. (1978) Topology, Allyn and Bacon Series in Advanced Mathematics, Boston, MA, Allyn and Bacon Inc., reprinting of the 1966 original.
Eells, J. and Elworthy, K. D. (1970) Open embeddings of certain Banach manifolds, Ann. of Math. (2) 91, 465–485.
Floer, A. (1988)a Morse theory for Lagrangian intersections, J. Differential Geom. 28, 513–547.
Floer, A. (1988)b A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41, 393–407.
Floer, A. (1988)c The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775–813.
Floer, A. (1989) Witten’s complex and infinite-dimensional Morse theory, J. Diifferential Geom. 30, 207–221.
Franks, J. M. (1979) Morse–Smale flows and homotopy theory, Topology 18, 199–215.
Franks, J. M. (1980) Homology and Dynamical Systems, Vol. 49 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.
Geba, K., Izydorek, M., and Pruszko, A. (1999) The Conley index in Hilbert spaces and its applications, Studia Math. 134, 217–233.
Guest, M. A. (1997) Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38 ofLondon Math. Soc. Stud. Texts, Cambridge, Cambridge Univ. Press.
Harvey, F. R. and Lawson, Jr., H. B. (2001) Finite volume flows and Morse theory, Ann. of Math. (2) 153, 1–25.
Izydorek, M. (2001) A cohomological Conley index in Hilbert spaces and applications to strongly indefinite problems, J. Differential Equations 170, 22–50.
Kato, T. (1980) Perturbation Theory for Linear Operators, Berlin, Springer, 2 edition.
Klingenberg, W. (1978) Lectures on Closed Geodesics, Vol. 230 of Grundlehren Math. Wiss., Berlin, Springer.
Klingenberg, W. (1982) Riemannian Geometry, Vol. 1 of de Gruyter Stud. Math., Berlin, de Gruyter.
Kryszewski, W. and Szulkin, A. (1997) An infinite-dimensional Morse theory with applications, Trans. Amer. Math. Soc. 349, 3181–3234.
Kuiper, N. H. (1965) The homotopy type of the unitary group of Hilbert space, Topology 3, 19–30.
Lang, S. (1999) Fundamentals of Differential Geometry, Vol. 191 of Grad. Texts in Math., New York, Springer.
Lasry, J.-M. and Lions, P.-L. (1986) A remark on regularization in Hilbert spaces, Israel J. Math. 55, 257–266.
Mawhin, J. and Willem, M. (1989) Critical Point Theory and Hamiltonian Systems, Vol. 74 of Appl. Math. Sci., New York, Springer.
Milnor, J. (1963) Morse Theory, Vol. 51 of Ann. of Math. Stud., Princeton, NJ, Princeton Univ. Press.
Milnor, J. W. (1965) Topology from the Differentiable Viewpoint, Charlottesville, VA, The University Press of Virginia.
Morse, M. (1925) Relations between the critical points of a real function of n independent variables, Trans. Amer. Math. Soc. 27, 345–396.
Morse, M. (1934) The Calculus of Variations in the Large, Vol. 18 of Amer. Math. Soc. Colloq. Publ, Providence, RI, Amer. Math. Soc.
Morse, M. (1947) Introduction to Analysis in the Large, Princeton, NJ, Princeton Univ. Press.
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.
Palais, R. S. (1963) Morse theory on Hilbert manifolds, Topology, 2 299–340.
Palais, R. S. (1965) On the homotopy type of certain groups of operators, Topology 3, 271–279.
Palis, J. (1968) On Morse–Smale dynamical systems, Topology 8, 385–405.
Palis, Jr., J. and de Melo, W. (1982) Geometric Theory of Dynamical Systems, New York, Springer.
Poźniak, M. (1991) The Morse complex, Novikov cohomology and Fredholm theory, preprint, University of Warwick.
Pressley, A. and Segal, G. (1986) Loop Groups, Oxford Math. Monogr., Oxford, Oxford Univ. Press.
Quillen, D. (1985) Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl. 19, 31–34.
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.
Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1–41.
Salamon, D. (1990) Morse theory, the Conley index and Floer homology, Bull. Amer. Math. Soc. 22, 113–140.
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.
Sato, M. (1981) Soliton equations as dynamical systems on an infinite-dimensional Grassman manifold, Sūrikaisekikenkyūsho Kōkyūroku 439, 30–46.
Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Mah, Basel, Birkhäuser.
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.
Segal, G. and Wilson, G. (1985) Loop groups and equations of KdV type, Inst. Hautes Études Sci. Publ. Math. 61, 5–65.
Shub, M. (1987) Global Stability of Dynamical Systems, New York, Springer.
Smale, S. (1960) Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66, 43–49.
Smale, S. (1961) On gradient dynamical systems, Ann. of Math.(2) 74, 199–206.
Smale, S. (1964)a A generalized Morse theory, Bull. Amer. Math. Soc. 70, 165–172.
Smale, S. (1964)b Morse theory and a nonlinear generalization of the Dirichlet problem, Ann. of Math.(2) 80, 382–396.
Smale, S. (1965) An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, 861–866.
Smale, S. (1967) Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747–817.
Szulkin, A. (1992) Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, 375–418.
Thom, R. (1949) Sur une partition en cellules associée à une fonction sur une variété, C. R. Acad. Sci. Paris 228, 973–975.
Weber, J. (1993) Der Morse–Witten Komplex, Master’s thesis, TU Berlin.
Weber, J. (2005) The Morse–Witten complex via dynamical systems, Exposition. Math., to appear.
Witten, E. (1982) Supersymmetry and Morse theory, J. Dierential Geom. 17, 661–692.
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ABBONDANDOLO, A., MAJER, P. (2006). LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS. In: Biran, P., Cornea, O., Lalonde, F. (eds) Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology. NATO Science Series II: Mathematics, Physics and Chemistry, vol 217. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4266-3_01
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