Connected ω-limit sets of Cournot maps: periodic orbits, realization and almost every point
Abstract
Let F(x,y)=(f2(y),f1(x)) be a Cournot map of the unit square, φ=f2∘ f1 and χ=f1∘ f2. 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)} ωF2(z)∩ωF2(F(z))≠∅, and conjectured that this holds for every ω-limit set. We prove the conjecture for every point z=(x,y) such that ωφ(x) or ωχ(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× J)∪(K× Q) with P⊂ K, Q⊂ 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.
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BibTeX
@misc{guirao2026connected,
title = {{Connected $ω$-limit sets of Cournot maps: periodic orbits, realization and almost every point}},
author = {Juan Luis García Guirao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27229},
eprint = {2610.27229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27229},
}
RIS
TY - UNPB
TI - Connected ω-limit sets of Cournot maps: periodic orbits, realization and almost every point
AU - Juan Luis García Guirao
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.27229
UR - https://arxiv.org/abs/2610.27229
AB - Let F(x,y)=(f_2(y),f_1(x)) be a Cournot map of the unit square, φ=f_2∘ f_1 and χ=f_1∘ 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)∩ω_F^2(F(z))≠∅, 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× J)∪(K× Q) with P⊂ K, Q⊂ 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.
ER -