Juan Luis García Guirao

arXiv2026

$ω$-limit sets of triangular maps over periodic orbits of the base

Let $F(x,y)=(f(x),g(x,y))$ be a continuous triangular map of the unit square and let $z=(x_0,y_0)$ be a point such that $ω_f(x_0)$ is a periodic orbit $\{p_0,\dots,p_{m-1}\}$ of $f$. Then $ω_F(z)=\bigcup_{i}\{p_i\}\times M_i$ for nonempty compact sets $M_i$, and we ask which $m$-tuples $(M_0,\dots,M_{m-1})$ occur. For $m=1$ the answer is a theorem of Kolyada and Snoha. We show that a tuple occurs if and only if there are continuous surjections $M_i\to M_{i+1}$, indices modulo $m$, whose composition around the cycle is chain transitive. This is a cyclic version of classical facts due to Dowker and Friedlander, Sharkovsky, Hirsch, Smith and Zhao, and Kolyada and Snoha. It yields two necessary conditions: no $M_i$ is one of the exceptional sets of Kolyada and Snoha, and there are continuous surjections $M_0\to M_1\to\dots\to M_{m-1}\to M_0$. We prove that these conditions are also sufficient when every $M_i$ has uncountably many connected components, and when every $M_i$ has countably many components of which only finitely many are nondegenerate. In particular the sections of one $ω$-limit set need not be homeomorphic. Every tuple satisfying the two conditions falls into one of these cases or into a single remaining class, in which the sections have countably many components and infinitely many of them are nondegenerate; that class is studied in a companion paper.

math.DS↗

arXiv2026

Connected $ω$-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, $φ=f_2\circ f_1$ and $χ=f_1\circ f_2$. Linero Bas and Muñoz Guillermo proved that an $ω$-limit set of $F$ with nonempty interior is connected if and only if \textup{(C1)} its two projections are connected and \textup{(C2)} $ω_{F^2}(z)\capω_{F^2}(F(z))\neq\emptyset$, and conjectured that this holds for every $ω$-limit set. We prove the conjecture for every point $z=(x,y)$ such that $ω_φ(x)$ or $ω_χ(y)$ is finite, and for Lebesgue almost every point $z$ of the square whenever $φ$ 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 $ω$-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 $\#P=\#Q$, or an \emph{alternating comb} of $2m$ disjoint segments. Conversely, every lattice and every alternating comb is an $ω$-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 $ω$-limit sets of Cournot maps. For instance, the letter H is not an $ω$-limit set at any such point, and for metrically regular $φ$ the points whose $ω$-limit set is connected, infinite, nowhere dense and not a lattice form a Lebesgue null set.

math.DS↗