$\omega $-Limit sets for triangular mappings

Victor Jiménez López ¹   ; Jaroslav Smítal ²

¹ Departamento de Matemáticas Universidad de Murcia Campus de Espinardo 30100 Murcia, Spain
² Institute of Mathematics Silesian University 746 01 Opava, Czech Republic

Fundamenta Mathematicae, Tome 167 (2001) no. 1, pp. 1-15

Résumé

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.

Détail

DOI : 10.4064/fm167-1-1

Keywords: agronsky ceder proved finite collection non degenerate peano continua unit square omega limit set continuous map improve result showing valid natural restrictions triangular maps mapsto square example non trivial peano continuum subset orbit enclosing omega limit set triangular map only has projection property finite union peano continua addition coherence property needed provide examples slightly different non peano continua square omega limit set triangular map view these examples characterization continua which omega limit sets triangular mappings seems difficult