Giraldo, A. and Rodríguez Sanjurjo, José Manuel (1999) Generalized Bebutov systems: a dynamical interpretation of shape. Journal of the mathematical society of japan, 51 (4). pp. 937-954. ISSN 0025-5645
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We define a semidynamical system - inspired by some classical dynamical systems studied by Bebutov in function spaces - in the space of approximative maps A(X, Y) between two metric compacta, with a suitable metric. Shape and strong shape morphisms are characterized as invariant subsets of this system. We study their structure and asymptotic properties and use the obtained results to give dynamical characterizations of basic notions in shape theory, like trivial shape, shape domination by polyhedra and internal FANRs.
|Uncontrolled Keywords:||Shape, shape morphisms, approximative maps, Bebutov semidynamical system|
|Subjects:||Sciences > Mathematics > Geometry|
Sciences > Mathematics > Topology
M. V. Bebutov, Sur les systemes dynamiques dans l’espace des fonctions continues, Doklady AN SSSR 27 (1940), 904–906 (see Mathematical Reviews 2, 225).
M. V. Bebutov, On dynamical systems in the space of continuous functions, Byulletin Moskovsk. Un-ta (Matematika) 2 (1941), 1–52 (in Russian).
N. P. Bhatia, Attraction and nonsaddle sets in dynamical systems, Journal of Di¤. Eq. 8 (1970), 229–249.
N. P. Bhatia and O. Hajek, Local semidynamical systems, Lecture Notes in Math. vol. 90 (Springer-Verlag, Berlin, 1969).
N. P. Bhatia and G. P. Szego, Stability theory of dynamical systems, Grundlehren der Math. Wiss. vol. 161 (Springer-Verlag, Berlin, 1970).
S. A. Bogatyi and V. I. Gutsu, On the structure of attracting compacta, Di¤erentsial’nye Uravneniya 25 (1989), 907–909 (in Russian).
K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223–254.
K. Borsuk, Theory of shape, Monografie Matematyczne vol. 59 (Polish Scientific Publishers, Warszawa, 1975).
T. A. Chapman, Lectures on Hilbert cube manifolds, Regional Conference Series in Math. vol. 28 (Amer. Math. Soc., 1976).
J. M. Cordier and T. Porter, Shape theory. Categorical methods of approximation, Ellis Horwood Series: Mathematics and its Applications (Ellis Horwood Ltd, Chichester, 1989).
J. Dydak, On internally movable compacta, Bull. Acad. Polon. Sci. 27 (1979), 107–110.
J. Dydak and J. Segal, Shape theory: An introduction, Lecture Notes in Math. vol. 688 (Springer-Verlag, Berlin, 1978).
J. Dydak and J. Segal, Strong shape theory, Dissertationes Math. vol. 192 (1981).
D. A. Edwards and H. M. Hastings, C ˇ ech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math. vol. 542 (Springer-Verlag, Berlin, 1976).
B. M. Garay, Strong cellularity and global asymptotic stability, Fund. Math. 138 (1991), 147–154.
A. Giraldo and J. M. R. Sanjurjo, Strong multihomotopy and Steenrod loop spaces, J. Math. Soc. Japan 47 (1995), 475–489.
A. Giraldo and J. M. R. Sanjurjo, On the global structure of invariant regions of flows with asymptotically stable attractors, Preprint.
B. Gu¨nther and 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.
H. M. Hastings, Shape theory and dynamical systems, In The structure of attractors in dynamical systems (N. G. Markley and W. Perizzo, eds.), Lecture Notes in Math. vol. 668 (Springer-Verlag, Berlin, 1978), pp. 150–160.
Y. Kodama and J. Ono, On fine shape theory I, Fund. Math. 105 (1979), 29–39.
Y. Kodama and J. Ono, On fine shape theory II, Fund. Math. 108 (1980), 89–98.
V. Laguna and J. M. R. Sanjurjo, Internal fundamental sequences and approximative retracts, Top. and Appl. 17 (1984), 189–197.
V. F. Laguna and J. M. R. Sanjurjo, Spaces of approximative maps, Math. Japonica 31 (1986), 623–633.
V. F. Laguna and J. M. R. Sanjurjo, Shape morphisms and spaces of approximative maps, Fund. Math. 133 (1989), 225–235.
S. Mardesic and J. Segal, Shape theory. (North Holland, Amsterdam, 1982).
T. Porter, Cech homotopy I, J. London Math. Soc. 6 (1973), 429–436.
J. B. Quigley, An exact sequence from the nth to (n-1)st fundamental group, Fund. Math. 77 (1973), 195–210.
J. W. Robbin and D. Salamon, Dynamical systems, shape theory and the Conley index, Ergod. Th. & Dynam. Sys. 8* (1988), 375–393.
J. T. Rogers, Jr., The shape of a cross-section of the solution funnel of an ordinary differential equation, Illinois J. Math. 21 (1977), 420–426.
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. (3) 69 (1994), 330–344.
J. M. R. Sanjurjo, On the structure of uniform attractors, Journal of Math. Analysis and its Applications 152 (1995), 519–528.
P. Saperstone, Semidynamical systems in infinite dimensional spaces, Applied Math. Sciences vol. 37 (Springer-Verlag, Berlin, 1981)
K. S. Sibirsky, Introduction to topological dynamics. (Noordhoff International Publishing, Leyden, 1975).
C. Tezer, Shift equivalence in homotopy, Math. Z. 210 (1992), 197–201
|Deposited On:||26 Oct 2012 09:44|
|Last Modified:||07 Feb 2014 09:37|
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