Generalizations of a theorem of Šarkovskii on orbits of continuous real-valued functions
type: Item
instance of: scholarly article
title: Generalizations of a theorem of Šarkovskii on orbits of continuous real-valued functions
zbMATH Open document ID: 0632.06005
DOI: 10.1016/0012-365X(87)90021-5
published in: Discrete Mathematics
publication date: 1987-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 4027527
zbMATH Keywords: partial ordering
zbMATH Keywords: point of least period
zbMATH Keywords: Sarkovskii ordering
zbMATH Keywords: cycles
zbMATH Keywords: orbit
zbMATH Keywords: combinatorial algorithm
author: Stewart Baldwin
MaRDI profile type: Publication
cites work: Simple Periodic Orbits of Mappings of the Interval
cites work: Stratification of Continuous Maps of an Interval
cites work: Q3893024
cites work: Q4745027
cites work: Q3042053
cites work: On the abundance of aperiodic behaviour for maps on the interval
cites work: On finite limit sets for transformations on the unit interval
cites work: Q5734070
cites work: A theorem of Sarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line
Recommended article: Unimodal orbits in continuous functions and generalization of Sarkovskii's theorem
Recommended article: Q3562047
Recommended article: Q3338932
Recommended article: Q3728704
Recommended article: Q5324098
Recommended article: Q3155805
Recommended article: Q3833264
Recommended article: Q5464380
Recommended article: Orbits of Darboux-like real functions
Recommended article: A complete realization of the orbits of generalized derivatives of quasiregular mappings
description: scientific article; zbMATH DE number 4027527