Hausdorff compactness on a space of ω-limit sets

摘要

In [Trans. Amer. Math. Soc. 348 (4) (1996)], Blokh et al, studied the space of the ω-limit sets W(f), produced by a continuous map on I = [0,1], and established that endowed with the Hausdorff metric topology on I, this space is compact. For general continuous maps on I~2, we show that this space is not compact and for maps whose W(F) is included in a fiber of I~2, we present examples of both types: holding and not holding the property of being compact.