Giraldo, A. and Morón, Manuel A. and Romero Ruiz del Portal, Francisco and Rodríguez Sanjurjo, José Manuel (2001) Some duality properties of non-saddle sets. Topology and its Applications, 113 (1-3). pp. 51-59. ISSN 0166-8641
Restricted to Repository staff only until 31 December 2020.
We show in this paper that the class of compacts that call be isolated non-saddle sets of flows in ANRs is precisely the class of compacta with polyhedral shape. We also prove-reinforcing the essential role played by shape theory in this setting-that the Conley index of a regular isolated non-saddle set is determined, in certain cases, by its shape. We finally introduce and study the notion of dual of a non-saddle set. Examples of compacta related by duality are attractor-repeller pairs. We use the complement theorems in shape theory to prove that the shape of the dual set is determined by the shape of the original non-saddle set.
|Uncontrolled Keywords:||Dynamical system; isolated set; non-saddle set; shape|
|Subjects:||Sciences > Mathematics > Topology|
A. Beck, On invariant sets, Ann. of Math. (2) 67 (1958) 99–103.
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, Grundlehren der Math. Wiss.,Vol. 161, Springer, Berlin, 1970.
S.A. Bogatyi, V.I. Gutsu, On the structure of attracting compacta, Differentsial’nye Uravneniya 25 (1989) 907–909 (in Russian).
K. Borsuk, Theory of Shape, Monografie Matematyczne, Vol. 59, Polish Scientific Publishers, Warszawa, 1975.
C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., Vol. 38, Amer. Math. Soc., Providence, RI, 1976.
J.M. Cordier, T. Porter, Shape Theory. Categorical Methods of Approximation, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1989.
R.J. Daverman, Decompositions of Manifolds, Academic Press, New York, 1986.
J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., Vol. 688, Springer, Berlin, 1978.
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.
D.M. Hyman, ANR divisors and absolute neighborhood contractibility, Fund. Math. 62 (1968) 61–73.
I. Ivanšić, R.B. Sher, A complement theorem for continua in a manifold, Topology Proc. 4 (1979) 437–452.
I. Ivanšić, R.B. Sher, G.A. Venema, Complement theorems beyond the trivial range, Illinois J. Math. 25 (1981) 209–220.
S. Mardešić, Pairs of compacta and trivial shape, Trans. Amer. Math. Soc. 189 (1974) 329–336.
S. Mardešić, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
J.W. Robbin, D. Salamon, Dynamical systems, shape theory and the Conley index, Ergod. Theory Dynamical Systems 8∗ (1988) 375–393.
J.M.R. Sanjurjo, Multihomotopy, Čech spaces of loops and shape groups, Proc. London Math. Soc. (3) 69 (1994) 330–344.
R.B. Sher, Complement theorems in shape theory, in: S. Mardešić, J. Segal (Eds.), Shape Theory and Geometric Topology, Lecture Notes in Math., Vol. 870, Springer, Berlin, 1981.
J.E. West, Mapping Hilbert cube manifolds to ANR’s: A solution of a conjecture of Borsuk, Ann. Math. 106 (1977) 1–18.
|Deposited On:||29 May 2012 11:35|
|Last Modified:||06 Feb 2014 10:24|
Repository Staff Only: item control page