We discuss the ideas of Morse decompositions and index filtrations for isolated invariant sets for both single-valued and multi-valued maps. We introduce the definition of connection matrix pairs and present the theorem of their existence. Connection matrix pair theory for multi-valued maps is used to show that connection matrix pairs obey the continuation property. We conclude by addressing applications to numerical analysis. This paper is primarily an overview of the papers [R1] and [R2].
@article{bwmeta1.element.bwnjournal-article-bcpv47i1p219bwm,
author = {Richeson, David},
title = {Connection matrix pairs},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {219-232},
zbl = {0946.37006},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p219bwm}
}
Richeson, David. Connection matrix pairs. Banach Center Publications, Tome 50 (1999) pp. 219-232. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv47i1p219bwm/
[000] [C] C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. in Math. 38, AMS, Providence, 1978.
[001] [Fr] J. Franks, Homology and Dynamical Systems, CBMS Reg. Conf. Ser. in Math. 49, AMS, Providence, 1982.
[002] [F1] R. Franzosa, The Connection Matrix Theory for Morse Decompositions, Trans. AMS 311 (1989), 561-592. | Zbl 0689.58030
[003] [F2] R. Franzosa, The Continuation Theory For Morse Decompositions and Connection Matrices, Trans. AMS 310 (1988), 781-803. | Zbl 0708.58021
[004] [FM] R. Franzosa and K. Mischaikow, The Connection Matrix Theory for Semiflows on (Not Necessarily Locally Compact) Metric Spaces, J. Differential Equations 71 (1988), 270-287. | Zbl 0676.54048
[005] [G1] L. Górniewicz, Homological Methods in Fixed Point Theory of Multi-valued Maps, Dissertationes Mathematicae 129 (1976). | Zbl 0324.55002
[006] [G2] L. Górniewicz, Topological Degree of Morphisms and its Applications to Differential Inclusions, Raccolta di Seminari del Dipartimento di Matematica dell'Università degli Studi della Calabria 5 (1983).
[007] [KM1] T. Kaczynski and M. Mrozek, Conley Index for Discrete Multi-valued Dynamical Systems, Top. App. 65 (1995), 83-96. | Zbl 0843.54042
[008] [KM2] T. Kaczynski and M. Mrozek, Stable Index Pairs for Discrete Dynamical Systems, preprint, 1994.
[009] [R1] D. Richeson, Connection Matrix Pairs for the Discrete Conley Index, preprint.
[010] [R2] D. Richeson, Morse Decompositions and Connection Matrix Pairs For Multi-valued Maps, preprint.