Let be a sequence of continuous interval maps which converges uniformly to a continuous map f. We study the limit behaviour of sequences with the form , , and whether the simplicity (respectively chaoticity) of f implies the simplicity (respectively chaoticity) of .

Given two continuous interval maps f and g, we study the periodic structure for the set of sequences generated by f and g by the rule . It is shown that this structure is intimately related to that of Sharkovsky's Theorem, although some differences appear with the periods that are coprime with 2.

We consider sequences of continuous maps defined on a compact metric space which uniformly converges to a continuous map f. We study the relationship between the ω-limit sets of f _1,∞ and f.