The continuation theory for Morse decompositions and connection matrices

Franzosa, Robert D.

doi:10.1090/S0002-9947-1988-0973177-6

The continuation theory for Morse decompositions and connection matrices
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by Robert D. Franzosa

Trans. Amer. Math. Soc. 310 (1988), 781-803

DOI: https://doi.org/10.1090/S0002-9947-1988-0973177-6

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Abstract:

The continuation theory for ($<$-ordered) Morse decompositions and the indices defined on them—the homology index braid and the connection matrices—is established. The equivalence between $<$-ordered Morse decompositions and $<$-consistent attractor filtrations is displayed. The spaces of ($<$-ordered) Morse decompositions for a product parametrization of a local flow are introduced, and the local continuation of ($<$-ordered) Morse decompositions is obtained via the above-described equivalence and the local continuation of attractors. The homology index braid and the connection matrices of an admissible ordering of a Morse decomposition are shown to be invariant on path components of the corresponding space of $<$-ordered Morse decompositions. This invariance is used to prove that the collection of connection matrices of a Morse decomposition is upper semicontinuous over the space of Morse decompositions (and over the parameter space) under local continuation.

References

Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
Charles Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984), no. 2, 207–253. MR 733717, DOI 10.1002/cpa.3160370204

Index filtrations and connection matrices for partially ordered Morse decompositions

Robert Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193–213. MR 857439, DOI 10.1090/S0002-9947-1986-0857439-7
Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561–592. MR 978368, DOI 10.1090/S0002-9947-1989-0978368-7
Robert D. Franzosa and Konstantin Mischaikow, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Differential Equations 71 (1988), no. 2, 270–287. MR 927003, DOI 10.1016/0022-0396(88)90028-9

Connections in parameterized families of flows via similarity transformations on connection matrices

Henry L. Kurland, The Morse index of an isolated invariant set is a connected simple system, J. Differential Equations 42 (1981), no. 2, 234–259. MR 641650, DOI 10.1016/0022-0396(81)90028-0

Homotopy invariants of repeller-attractor pairs

The Püppe sequence of an

pair

46

Homotopy invariants of repeller-attractor pairs

Continuation of

pairs

49

Mappings and homological properties in the Conley index theory

Classification of traveling wave solutions of reaction-diffusion systems

Homoclinic orbits in Hamiltonian systems and heteroclinic orbits in gradient and gradient-like systems

Existence of generalized homoclinic orbits for one parameter families of flows

The connection matrix and the classification of flows arising from ecological models

Connecting orbits in one parameter families of flows

Travelling wave solutions to a gradient system

Dietmar Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044, DOI 10.1090/S0002-9947-1985-0797044-3
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112