Geodesic
The omega limit sets of subsets in a metric space
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 87-96
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega $-limit set $\omega (Y)$ of $Y$ is the limit point of the sequence $\lbrace (\mathop {\mathrm Cl}Y)\cdot [i,\infty )\rbrace _{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.
In this paper, we discuss the properties of limit sets of subsets and attractors in a compact metric space. It is shown that the $\omega $-limit set $\omega (Y)$ of $Y$ is the limit point of the sequence $\lbrace (\mathop {\mathrm Cl}Y)\cdot [i,\infty )\rbrace _{i=1}^{\infty }$ in $2^X$ and also a quasi-attractor is the limit point of attractors with respect to the Hausdorff metric. It is shown that if a component of an attractor is not an attractor, then it must be a real quasi-attractor.
Classification : 34C35, 37B25, 37B30, 37C10, 54H20
Keywords: limit set of a set; attractor; quasi-attractor; hyperspace 
@article{CMJ_2005_55_1_a4,
     author = {Ding, Changming},
     title = {The omega limit sets of subsets in a metric space},
     journal = {Czechoslovak Mathematical Journal},
     pages = {87--96},
     year = {2005},
     volume = {55},
     number = {1},
     mrnumber = {2121657},
     zbl = {1081.37001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2005_55_1_a4/}
}
TY  - JOUR
AU  - Ding, Changming
TI  - The omega limit sets of subsets in a metric space
JO  - Czechoslovak Mathematical Journal
PY  - 2005
SP  - 87
EP  - 96
VL  - 55
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/CMJ_2005_55_1_a4/
LA  - en
ID  - CMJ_2005_55_1_a4
ER  - 
%0 Journal Article
%A Ding, Changming
%T The omega limit sets of subsets in a metric space
%J Czechoslovak Mathematical Journal
%D 2005
%P 87-96
%V 55
%N 1
%U http://geodesic.mathdoc.fr/item/CMJ_2005_55_1_a4/
%G en
%F CMJ_2005_55_1_a4
Ding, Changming. The omega limit sets of subsets in a metric space. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 87-96. http://geodesic.mathdoc.fr/item/CMJ_2005_55_1_a4/

[1] N. P. Bhatia and G. P. Szegö: Stability Theory of Dynamical Systems. Springer-Verlag, Berlin, 1970. | MR

[2] G. Butler and P. Waltmann: Persistence in dynamical systems. J.  Differential Equations 63 (1986), 255–263. | DOI | MR

[3] C. C. Conley: The gradient structure of a flow: I. Ergod. Th. & Dynam. Sys. $8^*$ (1988), 11–26. | DOI | MR | Zbl

[4] C. C. Conley: Isolated invariant sets and Morse index. Conf. Board Math. Sci., No  38, Amer. Math. Sci., Providence, 1978. | MR

[5] C. C. Conley: Some abstract properties of the set of invariant sets of a flow. Illinois J.  Math. 16 (1972), 663–668. | DOI | MR | Zbl

[6] J. K. Hale and P. Waltmann: Persistence in infinite-dimensional systems. SIAM J.  Math. Anal. 20 (1989), 388–395. | DOI | MR

[7] R. Moeckel: Some comments on “The gradient structure of a flow: I”. vol. $8^*$, Ergod. Th. & Dynam. Sys., 1988. | MR

[8] S. B. Nadler, Jr.: Continuum Theory: An Introduction. Marcel Dekker, New York-Basel-Hong Kong, 1992. | MR | Zbl

[9] T. Huang: Some global properties in dynamical systems. PhD. thesis, Inst. of Math., Academia Sinica, , 1998.