ON THE ω-LIMIT SETS OF PRODUCT MAPS

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, . . . , f_m, the union of all ω-limit sets of the product map f₁ × · · · × fm and the cartesian product of the sets ω(f₁), . . . , ω(f_m) coincide. This result enriches the theory of multidimensional permutation product maps, i.e., maps of the form F( x₁, . . . , x_m ) = ( f_σ(1) (x_σ(1)), . . . , f_σ(m)(x_σ(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.].