ω-limit sets of triangular maps over periodic orbits of the base
Abstract
Let F(x,y)=(f(x),g(x,y)) be a continuous triangular map of the unit square and let z=(x0,y0) be a point such that ωf(x0) is a periodic orbit {p0,…,pm−1} of f. Then ωF(z)=⋃i{pi}× Mi for nonempty compact sets Mi, and we ask which m-tuples (M0,…,Mm−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 Mi→ Mi+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 Mi is one of the exceptional sets of Kolyada and Snoha, and there are continuous surjections M0→ M1→…→ Mm−1→ M0. We prove that these conditions are also sufficient when every Mi has uncountably many connected components, and when every Mi 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.
Related
- Connected ω-limit sets of Cournot maps: periodic orbits, realization and almost every point
- On the rank-one limit sets for maps in the constant-Jacobian shear group of ℂ2
- At most exponential growth of periodic orbits for star flows
- Lyapunov-Schmidt reduction for periodic orbits of implicit discrete dynamical systems
- Cardinalities of Jordan Diophantine sets of upper triangular integer matrices
BibTeX
@misc{guirao2026ωlimit,
title = {{$ω$-limit sets of triangular maps over periodic orbits of the base}},
author = {Juan Luis García Guirao},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06889},
eprint = {2611.06889},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06889},
}
RIS
TY - UNPB
TI - ω-limit sets of triangular maps over periodic orbits of the base
AU - Juan Luis García Guirao
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06889
UR - https://arxiv.org/abs/2611.06889
AB - 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,…,p_m−1} of f. Then ω_F(z)=⋃_i{p_i}× M_i for nonempty compact sets M_i, and we ask which m-tuples (M_0,…,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→ 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→ M_1→…→ M_m−1→ 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.
ER -