Artículos de revistas
Spectral Sequences In Conleys Theory
Registro en:
Ergodic Theory And Dynamical Systems. , v. 30, n. 4, p. 1009 - 1054, 2010.
1433857
10.1017/S0143385709000479
2-s2.0-77955854414
Autor
Cornea O.
De Rezende K.A.
Da Silveira M.R.
Institución
Resumen
In this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an action type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials d rp:ErpErpr. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating Ep and Epr in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications. © 2009 Cambridge University Press. 30 4 1009 1054 Barraud, J.F., Cornea, O., Lagrangian intersections and the Serre spectral sequence (2007) Ann. of Math. (2), 166, pp. 657-722 Bredon, G.E., (1993) Topology and Geometry (Graduate Texts in Mathematics 139), , Springer New York Conley, C., (1978) Isolated Invariant Sets and the Morse Index (CBMS Regional Conference Series in Mathematics, 38), , American Mathematical Society, Providence, RI Cornea, O., Homotopical dynamics: Suspension and duality (2000) Ergod. Th. & Dynam. Sys., 20, pp. 379-391 Cornea, O., Homotopical dynamics II: Hopf invariants, smoothing and the Morse complex (2002) Ann. Sci. École. Norm. Sup. (4), 35, pp. 549-573 Cornea, O., Homotopical dynamics IV: Hopf invariants and Hamiltonian flows (2002) Comm. Pure Appl. Math., 55, pp. 1033-1088 Cruz, R.N., De Rezende, K.A., Mello, M., Realizability of the Morse polytope (2007) Qual. Theory Dyn. Syst., 6, pp. 59-86 Davis, J.F., Kirk, P., (2001) Lecture Notes in Algebraic Topology (Graduate Studies in Mathematics, 35), , American Mathematical Society, Providence, RI Franks, J., Morse-Smale flows and homotopy theory (1979) Topology, 18, pp. 199-215 Franks, J., (1982) Homology*Dynamical Systems (CBMS Regional Conference Series in Mathematics 49), , American Mathematical Society*Providence RI Franzosa, R., Index filtrations and the homology index braid for partially ordered Morse decompositions (1986) Trans. Amer. Math. Soc., 298, pp. 193-213 Franzosa, R., The continuation theory for Morse decompositions and connection matrices (1988) Trans. Amer. Math. Soc., 310, pp. 781-803 Franzosa, R., The connection matrix theory for Morse decompositions (1989) Trans. Amer. Math. Soc., 311, pp. 561-592 Franzosa, R., Mischaikow, K., Algebraic transition matrices in the Conley index theory (1998) Trans. Amer. Math. Soc., 350, pp. 889-912 Kurland, H.L., Homotopy invariants of repeller-attractor pairs I: The Puppe sequence of an R-A pair (1982) J. Differential Equations, 46, pp. 1-31 Leclercq, R., Spectral Invariants in Lagrangian Floer Theory, , Preprint 2006 Available at arXiv:math/0612325 McCord, C., The connection map for attractor-repeller pairs (1988) Trans. Amer. Math. Soc., 307, pp. 195-203 McCord, C., Reineck, J.F., Connection matrices and transition matrices (1999) Conley Index Theory (Banach Center Publications 47), pp. 41-55. , Polish Academy of Sciences Warsaw Milnor, J.W., (1965) Topology from the Differentiable Viewpoint, , University Press of Virginia, Charlottesville, VA Milnor, J.W., (1965) Lectures on the H-Cobordism Theorem, , Princeton University Press, Princeton, NJ Moeckel, R., Morse decompositions and connection matrices (1988) Ergod. Th. & Dynam. Sys., 8, pp. 227-249 Reineck, J.F., The connection matrix in Morse-Smale flows (1990) Trans. Amer. Math. Soc., 322, pp. 523-545 Reineck, J.F., The connection matrix in Morse-Smale flows II (1995) Trans. Amer. Math. Soc., 347, pp. 2097-2110 Reineck, J.F., Continuation to the minimal number of critical points in gradient flows (1992) Duke Math. J., 68, pp. 185-194 Salamon, D., Connected simple systems and the Conley index of invariant sets (1985) Trans. Amer. Math. Soc., 291, pp. 1-41 Salamon, D., Morse theory, Conley index and Floer homology (1990) Bull. London Math. Soc., 22, pp. 113-140 Smale, S., The generalized Poincaré conjecture in higher dimensions (1960) Bull. Amer. Math. Soc., 66, pp. 373-375 Smale, S., On the structure of manifolds (1962) Amer. J. Math., 84, pp. 387-399 Spanier, E., (1966) Algebraic Topology, , McGraw-Hill, New York