A class of twisted products of maps of an interval
type: Item
instance of: scholarly article
title: A class of twisted products of maps of an interval
zbMATH Open document ID: 0811.58021
DOI: 10.1007/BF01209553
author: L. S. Efremova
published in: Mathematical Notes
publication date: 1994-11-16T00:00:00Z
review text: Let \(I\) be a closed rectangle in the plane with the product topology. The author discusses properties of \(C^ 1\)-smooth mappings \(F : I \to I\) being of the form \(F(x,y) = (f(x), g_ x(y))\) assuming some additional conditions (among them: \(\Omega\)-stability of \(g_ x\)). Dynamics of such maps over a quasiminimal set of the factor \(f\) and the structure of the set of periodic points of \(F\) and homoclinic points of \(F\) are investigated. In particular, it is proved (Theorem B) that for \(F\) being under consideration, the following three assertions are equivalent: (i) there exists a periodic point of period different from \(2^ i\), \(i = 0, 1, 2, \dots\) (ii) the set of periodic points is not a \(G_ \delta\)-set (iii) there is a homoclinic point.
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 687502
zbMATH Keywords: nonwandering set
zbMATH Keywords: quasiminimal set
zbMATH Keywords: periodic point
zbMATH Keywords: homoclinic point
reviewed by: Andrzej Pelczar
MaRDI profile type: Publication
cites work: Q4069541
cites work: On topological dynamics of triangular maps of the plane
cites work: Q5553902
cites work: Homoclinic Points of Mappings of the Interval
cites work: On Sharkovsky's cycle coexistence ordering
cites work: The chain recurrent set, attractors, and explosions
description: scientific article; zbMATH DE number 687502