Connected $\omega$-limit sets of Cournot maps: periodic orbits, realization and almost every point

Let $F(x,y)=(f_2(y),f_1(x))$ be a Cournot map of the unit square, $\phi =f_2\circ f_1$ and $\chi =f_1\circ f_2$. Linero Bas and Muñoz Guillermo proved that an $\omega$-limit set of $F$ with nonempty interior is connected if and only if \textup{(C1)} its two projections are connected and \textup{(C2)} ${\omega }_{F^2}(z)\cap {\omega }_{F^2}(F(z))\mathrm{\ne }\mathrm{\varnothing }$, and conjectured that this holds for every $\omega$-limit set. We prove the conjecture for every point $z=(x,y)$ such that ${\omega }_{\phi }(x)$ or ${\omega }_{\chi }(y)$ is finite, and for Lebesgue almost every point $z$ of the square whenever $\phi$ is metrically regular (every metric attractor is a periodic orbit, a solenoid, or a cycle of intervals carrying a well-behaved absolutely continuous invariant measure). Over a periodic orbit the $\omega$-limit set is a \emph{comb}, whose sections are rigid; when one projection is connected it is a point, a \emph{lattice} $(P\times J)\cup (K\times Q)$ with $P\subset K$, $Q\subset J$ and $\mathrm{\#}P=\mathrm{\#}Q$, or an \emph{alternating comb} of $2m$ disjoint segments. Conversely, every lattice and every alternating comb is an $\omega$-limit set of a Cournot map with piecewise linear reaction functions; the construction rests on a lemma producing exact compositions of piecewise linear maps with prescribed values. This answers, for this class, the question of Balibrea, Linero and Cánovas of which nowhere dense sets are $\omega$-limit sets of Cournot maps. For instance, the letter H is not an $\omega$-limit set at any such point, and for metrically regular $\phi$ the points whose $\omega$-limit set is connected, infinite, nowhere dense and not a lattice form a Lebesgue null set.