Topological Methods in Nonlinear Analysis

Connection matrices for Morse-Bott flows

Dahisy V. de S. Lima and Ketty A. de Rezende

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A Connection Matrix Theory approach is presented for Morse-Bott flows $\varphi$ on smooth closed $n$-manifolds by characterizing the set of connection matrices in terms of Morse-Smale perturbations. Further results are obtained on the effect on the set of connection matrices $\mathcal{CM}(S)$ caused by changes in the partial orderings and in the Morse decompositions of an isolated invariant set $S$.

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Topol. Methods Nonlinear Anal., Volume 44, Number 2 (2014), 471-495.

First available in Project Euclid: 11 April 2016

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Lima, Dahisy V. de S.; de Rezende, Ketty A. Connection matrices for Morse-Bott flows. Topol. Methods Nonlinear Anal. 44 (2014), no. 2, 471--495.

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  • A. Banyaga and D. Hurtubise, The Morse–Bott inequalities via a dynamical systems approach, Ergodic Theory Dynam. Systems 29 (2009), no. 6, 1693–1703.
  • ––––, Lecture on Morse Homology, Kluwer Texts in the Mathematical Sciences, vol. 29, Kluwer Academic Publishers Group, Dordrecht, (2004).
  • C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conference Series in Math. AMS, Providence, RI 38 (1978).
  • O. Cornea, K.A. de Rezende and M.R. da Silveira, Spectral sequences in Conley's theory, Ergodic Theory Dynam. Systems 30 (4) (2010), 1009–1054.
  • R. Franzosa, Index filtrations and the homology index braid for partially ordered Morse fecompositions, Trans. Amer. Math. Soc. 298 (1986), 193–213.
  • ––––, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), 561–592.
  • ––––, The continuation theory for Morse decompositions and connection matrices, Trans. Amer. Math. Soc. 310 (1988), 781–803.
  • R. Franzosa, K.A. de Rezende and M.R. da Silveira, Continuation and bifurcation associated to the dynamical spectral sequence, Ergodic Theory Dynam. Systems, available on CJO 2013 doi:10.1017/etds.2013.29.
  • D. Lima and K. de Rezende, Generalized Morse–Bott Inequalities, preprint.
  • C. McCord and K. Mischaikow, Connected simple systems, transition matrices and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333 (1992), 397–422.
  • M.P. Mello, K.A. de Rezende and M.R. da Silveira, Conley's spectral sequences via the sweeping algorithm, Topology Appl. 157 (13) (2010), 2111–2130.
  • J. Reineck, Connecting orbits in one-parameter families of flows, Ergodic Theory Dynam. Systems 8 (1988), 359–374.
  • ––––, The connection matrix in Morse–Smale flows, Trans. Amer. Math. Soc. 322 (1990), 523–545.
  • D.A. Salamon, The Morse theory, the Conley index and the Floer homology, Bull. London Math. Soc. 22 (1990), 113–240.
  • J. Weber, The Morse–Witten complex via dynamical systems, Expo. Math. 24 (2006), 127–159.