In a recent paper we provide a characterization of triangular maps of the square, i.e., maps given by $F(x,y)=(f(x),g_x(y))$, satisfying condition (C1) that any chain recurrent point is periodic. For continuous maps of the interval, there is a list of 18 other conditions equivalent to (C1), including (C2) that there is no infinite $\omega$-limit set, (C3) that the set of periodic points is closed and (C4) that any almost periodic point is periodic, for instance. We provide an almost complete classification among these conditions and one open problem. The paper solves partially a~problem formulated in the eighties by A. N. Sharkovsky.
The mentioned open problem, the validity of (C4) $ \Rightarrow$ (C3), is related
to
the question whether some almost periodic point lies in the fibres over a $f$-minimal
set possessing an almost periodic point. We answered this question in positive for
triangular maps with nondecreasing fiber maps. Consequently, the classification is
completed for motonone triangular maps.