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 show that where C is a positive constant and , and where k is an odd natural number, are the unique -cycles having the form where f _j are continuous selfmaps of the interval , .

The aim of this paper is to give an account of some results recently obtained in Combinatorial Dynamics and apply them to get for k S 2 the periodic structure of delayed difference equations of the form x n = f ( x n m k ) on I and S 1 .