Dynamic Systems and Applications 19 (2010) 667-679 ON THE ω-LIMIT SETS OF PRODUCT MAPS
نویسنده
چکیده
Let ω(·) denote the union of all ω-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps f1, . . . , fm, the union of all ω-limit sets of the product map f1 × · · · × fm and the cartesian product of the sets ω(f1), . . . , ω(fm) coincide. This result enriches the theory of multidimensional permutation product maps, i.e., maps of the form F (x1, . . . , xm) = (fσ(1)(xσ(1)), . . . , fσ(m)(xσ(m))), where σ is a permutation of the set of indices {1, . . . ,m}. For any such map F , we prove that the set ω(F ) is closed and we also show that ω(F ) cannot be a proper subset of the center of the map F . These results solve open questions mentioned, e.g., in [F. Balibrea, J. S. Cánovas, A. Linero, New results on topological dynamics of antitriangular maps, Appl. Gen. Topol.]. AMS (MOS) Subject Classification. 37E05, 54H20, 37E99, 37B99
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