It is well-known that, for a continuous map j of the interval, the condition P1 j has zero topological entropy, is equivalent, e.g., to any of the following: P2 any w -limit set contains a unique minimal set; P3 the period of any cycle of j is a power of two; P4 any w -limit set either is a cycle or contains no cycle; P5 if wj ( x ) = wj2 ( x ), then wj ( x ) is a fixed point; P6 j has no homoclinic trajectory; P7 there is no countably infinite w -limit set; P8 trajectories of any two points are correlated; P9 there is no closed invariant subset A such that j m | A is topologically almost conjugate to the shift, for some m 3 1. In the paper we exhibit the relations between these properties in the class ( x,y ) ® ( f ( x ), gx ( y )) of triangular maps of the square. This contributes to the solution of a longstanding open problem of Sharkovsky.