Asymptotical periodicity for analytic triangular maps of type less than 2∞

We prove that if F is an analytic triangular map of type less than 2∞ in the Sharkovsky ordering, then all points are asymptotically periodic for F. The same is true if, instead of being analytic, F is just continuous but has the property that each fibre contains finitely many periodic points. Improving earlier counterexamples in Kolyada (1992) [16], and Balibrea et al. (2002) [3], we also show that this need not be the case when F is a C∞ map. Finally we remark that type less than 2∞ and closedness of periodic points are equivalent properties in the C1 setting for triangular maps.