On the Sharkovsky's problem concerning classification of triangular maps

It is well-known that, for a continuous map φ of the interval, the condition φ has zero topological entropy, is equivalent, e.g., to any of the following: trajectories of any points can be approximated by trajectories of closed connected periodic sets in the stronger or weaker sense, respectively; any ω-limit set contains a unique minimal set; the period of any cycle of φ is a power of two; any ω-limit set either is a cycle or contains no cycle; if ω_φ(ξ)=ω_φ²(ξ), then ω_φ (ξ) is a fixed point; φ has no homoclinic trajectory; there is no countably infinite ω-limit set; trajectories of any two points are correlated; there is no closed invariant subset A such that φ^m|A is topologically almost conjugate to the shift, for some m ≥ 1. We exhibit relations between these properties in the class (x,y) \mapsto (f(x),g_x(y)) of triangular maps of the square. This contributes to the solution of a longstanding open problem of Sharkovsky.