On a Conjecture of Agronsky and Ceder Concerning Orbit-Enclosing \(\omega\)-Limit Sets

Abstract

In 1991, Agronsky and Cedar made the following conjecture: A continuum \(K \subset E^{k}\) is an orbit-enclosing \(\omega\)-limit set if and only if it is arcwise connected. The main aim of this paper is to disprove this conjecture by giving an example of an orbit-enclosing \(\omega\)-limit set \(S\) in \(E^{2}\) (cf. Theorem 3 below) which is not arcwise connected. Moreover, we show that \(S\) can be chosen with non-empty interior, and the mapping \(F\), with respect to which \(S\) is an orbit-enclosing \(\omega\)-limit set can be chosen as a triangular map.