In 1992 Agronsky and Ceder proved that any finite collection of non-degenerate Peano continua in the unit square is an ω-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) ↦ (f(x),g(x,y)) of the square. For example, we show that a non-trivial Peano continuum C ⊂ I² is an orbit-enclosing ω-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 ω-limit set of a triangular map. In view of these examples a characterization of the continua which are ω-limit sets for triangular mappings seems to be difficult.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-1-1,
author = {Victor Jim\'enez L\'opez and Jaroslav Sm\'\i tal},
title = {$\omega$-Limit sets for triangular mappings},
journal = {Fundamenta Mathematicae},
volume = {167},
year = {2001},
pages = {1-15},
zbl = {0972.37012},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-1-1}
}
Victor Jiménez López; Jaroslav Smítal. ω-Limit sets for triangular mappings. Fundamenta Mathematicae, Tome 167 (2001) pp. 1-15. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm167-1-1/