On a problem concerning $\omega$-limit sets of triangular maps
in $I^3$

We show that there is a continuous triangular map $I^3\rightarrow I^3$ with $\omega (F)=\{0\}\times I^2=\omega_F(x,y,z)$ for any $(x,y,z)\in I^3$ such that $x\neq 0$. This map is of the form $F(x,y,z)=(f(x),g(x,y),h(x,z))$, where the maps $g(x, . )$ and $h(x, . )$ are non-decreasing. This solves a problem by F. Balibrea, L. Reich, and J. Sm\'{\i}tal.

Keywords. Triangular map, $\omega$-limit set.

MS 2000 Classification numbers. 37B99,37E99.