Giraldo, A. and Rodríguez Sanjurjo, José Manuel (2004) Morse-Smale equations of non-saddle decompositions. Topology and its Applications, 140 (1). pp. 69-80. ISSN 0166-8641
Restricted to Repository staff only until 31 December 2020.
The notion of a non-saddle decomposition of a compact ANR is introduced. This notion extends that of a an attractor-repeller pair. Some cohomological properties of non-saddle decompositions are studied. In particular, some inequalities in the spirit of the Morse-Smale equations for attractor-repeller pairs are obtained. These inequalities involve the ranks of the cohomological Conley index and also of a new cohomological invariant introduced here. The notion of a cyclic Morse decomposition is also introduced and it is proved that this kind of decomposition admits filtrations by non-saddle sets. Finally, these filtrations are used to obtain Morse-Smale equations that generalize those of a Morse decomposition.
International Conference on Topology and Its Applications, SEP 02-09, 2000, Ohrid, MACEDONIA
|Uncontrolled Keywords:||Dynamical system; Isolated set; Non-saddle set; Non-saddle filtration; Cyclic Morse decomposition; Shape|
|Subjects:||Sciences > Mathematics > Geometry|
Sciences > Mathematics > Topology
N.P. Bhatia, Attraction and nonsaddle sets in dynamical systems, J. Differential Equations 8 (1970) 229–249.
N.P. Bhatia, G.P. Szego, Stability Theory of Dynamical Systems, in: Grundlehren Math. Wiss., vol. 161, Springer, Berlin, 1970.
S.A. Bogatyi, V.I. Gutsu, On the structure of attracting compacta, Differentsialnye Uravneniya 25 (1989) 907–909 (in Russian).
K. Borsuk, Theory of Shape, in: Monografie Mat., vol. 59, Polish Scientific Publishers, Warszawa, 1975.
C.C. Conley, Isolated Invariant Sets and the Morse Index, in: CBMS Regional Conf. Ser. in Math., vol. 38, American Mathematical Society, Providence, RI, 1976.
C.C. Conley, The gradient structure of a flow, Ergodic Theory Dynamical Systems 8* (1988) 11–26.
C.C. Conley, R.W. Eastern, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971) 35–61.
J.M. Cordier, T. Porter, Shape Theory. Categorical Methods of Approximation, in: Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood, Chichester, 1989.
J. Dydak, J. Segal, Shape Theory: An Introduction, in: Lecture Notes in Math., vol. 688, Springer, Berlin, 1978.
A. Giraldo, J.M.R. Sanjurjo, On the global structure of invariant regions of flows with asymptotically stable attractors, Math. Z. 232 (1999) 739–746.
A. Giraldo, M.A. Morón, F.R. Ruíz del Portal, J.M.R. Sanjurjo, Some duality properties of non-saddle sets, Topology Appl. 113 (2001) 51–59.
M. Gobbino, Topological properties of attractors for dynamical systems, Topology 40 (2001) 279–298.
B. Günther, J. Segal, Every attractor of a flow on a manifold has the shape of a finite polyhedron, Proc. Amer. Math. Soc. 119 (1993) 321–329.
I. Ivanšic, R.B. Sher, A complement theorem for continua in a manifold, Topology Proc. 4 (1979) 437–452.
I. Ivanšic, R.B. Sher, G.A. Venema, Complement theorems beyond the trivial range, Illinois J. Math. 25 (1981) 209–220.
L. Kapitanski, I. Rodnianski, Shape and Morse theory of attractors, Comm. Pure Appl. Math. 53 (2000) 218–242.
S. Mardešic, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
J.W. Robbin, D. Salamon, Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynamical Systems 8* (1988) 375–393.
J.M.R. Sanjurjo, Multihomotopy, Cˇ ech spaces of loops and shape groups, Proc. London Math. Soc. 69 (3)(1994) 330–344.
J.E. West, Mapping Hilbert cube manifolds to ANR’s: A solution of a conjecture of Borsuk, Ann. Of Math. 106 (1977) 1–18.
|Deposited On:||23 Oct 2012 08:42|
|Last Modified:||07 Feb 2014 09:36|
Repository Staff Only: item control page