On Iterated Positive Schwarzian derivative Maps

The families of one-dimensional maps have been the object of many studies since [Sharkovsky, 1964], for instance in [Sharkovsky et al., 1997] or in [de Melo & van Strien, 1993], whose authors also studied this subject, we have two large surveys of the enormous effort made in the last decades in this field of work. We adopt the usual definitions as for instance symbolic dynamics or topological equivalence from those works. As a rule when we say that some definition is usual or well known the reader can easily find those subjects in any one of the cited books. Our goal is the study of `-modal families of maps on several parameters with the possibility of changing the sign of the Schwarzian derivative when we change one or more of these parameters. Here we begin this study by one family of unimodal maps. In this section we present our object of study, the unimodal family of maps fa,b on the interval I = [−1, 1], depending on two real parameters, a and b: