Generalizations of a theorem of Šarkovskii on orbits of continuous real-valued functions
A cycle of length \(n\) is any bijection \(p: \{1,2,\dots,n\}\to \{1,2,\dots,n\}\) such that \(1, p(1), p^ 2(1),\dots, p^{n-1}(1)\) are all distinct. Let \(p, q\) be cycles. Then define \(p\to q\) iff for every continuous \(f: \mathbb R\to \mathbb R\), if f has an orbit of type \(p\), then \(f\) also has an orbit of type \(q\). \(\to\) is a partial (but not linear) ordering and generalizes the idea of \textit{A. N. Sharkovskii}'s ordering \(\geq_ s\) [Ukr. Mat. Zh. 16, 61--71 (1964; Zbl 0122.17504)]. The author also introduces the ideas of labelled digraph and closed walk. Using these concepts, this paper offers a combinatorial algorithm which decides whether \(p\to q\). Some interesting theorems about this topic are obtained. Several examples and conjectures are also given in this paper.
scientific article; zbMATH DE number 4027527
scientific article