Triangular maps of the square

Let $F$ be a triangular map $(x,y) \mapsto (f(x),g(x,y))$ from the unit square $I^{2}$ into itself. We consider a list of properties of this map, such as (i) $F$ has zero topological entropy; (ii) period of any cycle of $F$ is a power of 2; (iii) $F$ has no homoclinic trajectory; (iv) $F|\ \UR(F)$ is non-chaotic in the sense of Li and Yorke; (v) $\UR(F) = \Rec(F)$; (vi) $F$ is not DC1; (vii) $F$ is not DC2; (viii) $F$ is not DC3; and some others. Here DC1--DC3 denote distributional chaos of type 1--3. It is well-known that these properties are not mutually equivalent in this case (in contradistinction to the case $C(I,I)$). This paper is a survey of known relations between the properties in the case of triangular maps, and in the case of trigangular maps which are non-decreasing on the fibres.