$ω$-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 $z=(x_0,y_0)$ 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.