A classification of triangular maps of the square

It is well-known that, for a continuous map $\varphi$ of the interval, the condition {P1} $\varphi$ has zero topological entropy, is equivalent, e.g., to any of the following: { P2} any $\omega $-limit set contains a unique minimal set; { P3} the period of any cycle of $\varphi$ is a power of two; { P4} any $\omega$-limit set either is a cycle or contains no cycle; {P5} if $\omega _\varphi(\xi)=\omega_{\varphi^2}(\xi)$, then $\omega_\varphi (\xi)$ is a fixed point; {P6} $\varphi $ has no homoclinic trajectory; {P7} there is no countably infinite $\omega$-limit set; {P8} trajectories of any two points are correlated; {P9} there is no closed invariant subset $A$ such that $\varphi ^m|A$ is topologically almost conjugate to the shift, for some $m\ge 1$. In the paper we exhibit the 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.