$\omega $-Limit sets for triangular mappings
In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an $\omega $-limit set for a continuous map. We improve this result by showing that it is valid, with natural restrictions, for the triangular maps $(x,y)\mapsto (f(x),g(x,y))$ of the square. For example, we show that a non-trivial Peano continuum $C\subset I^2$ is an orbit-enclosing $\omega $-limit set of a triangular map if and only if it has a projection property. If $C$ is a finite union of Peano continua then, in addition, a coherence property is needed. We also provide examples of two slightly different non-Peano continua $C$ and $D$ in the square such that $C$ is and $D$ is not an $\omega $-limit set of a triangular map. In view of these examples a characterization of the continua which are $\omega $-limit sets for triangular mappings seems to be difficult.