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Shift equivalence and the Conley index. (English) Zbl 0956.37010
Summary: We introduce filtration pairs for an isolated invariant set of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Last, we define partially ordered Morse decompositions and prove the existence of Morse set filtrations for such decompositions.

MSC:
37B30 Index theory for dynamical systems, Morse-Conley indices
37D15 Morse-Smale systems
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[1] Rufus Bowen and John Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2) 106 (1977), no. 1, 73 – 92. · Zbl 0375.58018
[2] Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. · Zbl 0397.34056
[3] Robert Easton, Isolating blocks and epsilon chains for maps, Phys. D 39 (1989), no. 1, 95 – 110. · Zbl 0696.58042
[4] Robert Franzosa, Index filtrations and the homology index braid for partially ordered Morse decompositions, Trans. Amer. Math. Soc. 298 (1986), no. 1, 193 – 213. · Zbl 0626.58013
[5] Robert D. Franzosa, The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311 (1989), no. 2, 561 – 592. · Zbl 0689.58030
[6] Marian Mrozek, Leray functor and cohomological Conley index for discrete dynamical systems, Trans. Amer. Math. Soc. 318 (1990), no. 1, 149 – 178. · Zbl 0686.58034
[7] D. Richeson, Connection matrix pairs for the discrete Conley index, Ph.D. Dissertation, Northwestern University (1998).
[8] Joel W. Robbin and Dietmar Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems 8* (1988), no. Charles Conley Memorial Issue, 375 – 393. · Zbl 0682.58040
[9] Joel W. Robbin and Dietmar A. Salamon, Lyapunov maps, simplicial complexes and the Stone functor, Ergodic Theory Dynam. Systems 12 (1992), no. 1, 153 – 183. · Zbl 0737.58033
[10] S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747 – 817. · Zbl 0202.55202
[11] A. Szymczak, The Conley index for discrete semidynamical systems, Topology Appl. 66 (1995), no. 3, 215 – 240. · Zbl 0840.34043
[12] R. F. Williams, Classification of one dimensional attractors, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 341 – 361.
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