Discrete dynamical system; interval map; product map; permutation product map; antitriangular map; $\omega$-limit set; solenoidal set; basic set; center

Let $\omega(\cdot)$ denote the union of all $\omega$-limit sets of a given map. As the main result of this paper we prove that, for given continuous interval maps $f_1,\ldots, f_m$, the set of $\omega$-limit points of the product map $f_1 \times \cdots \times f_m$ and the cartesian product of the sets $\omega(f_1),\ldots, \omega(f_m)$ coincide. This result substantially enriches the theory of multidimensional permutation product maps, i.e., maps of the form $F(x_1,\ldots, x_m) = (f_{\sigma(1)}(x_{\sigma(1)}), \ldots,f_{\sigma(m)}(x_{\sigma(m)}))$, where $\sigma$ is a permutation of the set of indices $\{1,\ldots,m\}$. Especially, for any such map $F$, we prove that the set $\omega(F)$ is closed and we also show that $\omega(F)$ 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. C\'{a}novas, A. Linero, {\em New results on topological dynamics of antitriangular maps\/}, Appl. Gen. Topol.].