Giraldo, A. and Rodríguez Sanjurjo, José Manuel (2009) Topological robustness of non-saddle sets. Topology and its Applications, 156 (11). pp. 1929-1936. ISSN 0166-8641
Restricted to Repository staff only until 31 December 2020.
We study in this paper preservation of dynamical and shape theoretical properties under Continuation for parametrized families of flows. We show that, although attractors continue. the same does not hold for non-saddle sets. However, when they continue, their shape is preserved in quite general settings, which include differentiable families of flows and regular non-saddle sets for general flows, not necessarily differentiable. We also Study how the continuation of a non-saddle set influences that of its dual non-saddle set.
|Additional Information:||Conference on General Topology in Honour of Peter Collins and Mike Reed, AUG 07-10, 2006, Oxford, ENGLAND|
|Uncontrolled Keywords:||Shape equivalence, Dynamical system, Isolated invariant set, Continuation of non-saddle sets|
|Subjects:||Sciences > Mathematics > Geometry|
Sciences > Mathematics > Topology
A. Beck, On invariant sets, Ann. of Math. (2) 67 (1958) 99–103.
N.P. Bhatia, G.P. Szego, Stability Theory of Dynamical Systems, Grundlehren Math. Wiss., vol. 161, Springer-Verlag, Berlin, 1970.
S.A. Bogatyi, V.I. Gutsu, On the structure of attracting compacta, Differ. Urav. 25 (1989) 907–909 (in Russian).
K. Borsuk, Theory of Shape, Monogr. Mat., vol. 59, Polish Scientific Publishers, Warszawa, 1975.
C.C. Conley, Isolated Invariant Sets and the Morse Index, CBMS Reg. Conf. Ser. Math., vol. 38, Amer. Math. Soc., Providence, RI, 1976.
C.C. Conley, The gradient structure of a flow, Ergodic Theory Dynam. Systems 8∗ (1988) 11–26.
C.C. Conley, R.W. Easton, 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, Ellis Horwood Ser. Math. Appl., Ellis Horwood Ltd, Chichester, 1989.
J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., vol. 688, Springer-Verlag, Berlin, 1978.
R.W. Easton, Geometric Methods for Discrete Dynamical Systems, Oxford Engrg. Sci. Ser., vol. 50, Oxford University Press, New York, 1998.
S. Eilenberg, N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, Princeton, 1952.
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.
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.
S. Mardešic, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000.
S. Mardešic, J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
D. Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985) 1–41.
J.M.R. Sanjurjo, An intrinsic description of shape, Trans. Amer. Math. Soc. 329 (1992) 625–636.
J.M.R. Sanjurjo, Multihomotopy, Cech spaces of loops and shape groups, Proc. London Math. Soc. 69 (1994) 330–344.
J.M.R. Sanjurjo, On the structure of uniform attractors, J. Math. Anal. Appl. 192 (1995) 519–528.
J.M.R. Sanjurjo, Lusternik–Schnirelmann category and Morse decompositions, Mathematika 47 (2000) 299–305.
J.M.R. Sanjurjo, Morse equations and unstable manifolds of isolated invariant sets, Nonlinearity 16 (2003) 1435–1448.
|Deposited On:||16 Oct 2012 11:31|
|Last Modified:||16 Oct 2012 11:31|
Repository Staff Only: item control page