The Feigenbaum functional equation and periodic points
type: Item
instance of: scholarly article
title: The Feigenbaum functional equation and periodic points
zbMATH Open document ID: 0593.39001
DOI: 10.1007/BF02189926
author: B. George
published in: Aequationes Mathematicae
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137127
review text: The behavior of solutions of the Feigenbaum functional equation \[ (1)\quad f(f(\lambda x))+\lambda f(x)=0,\quad \lambda =f(-1) \] is studied. The following result is, particularly, proved. Theorem. If f is an even unimodal solution of (1) with \(-1<f(1)<0\), then f has a unique fixed point, say \(x_ 1\), where \(x_ 1>0\), \(\lambda <x_ 1<f(\lambda)\). If f does not have a period 2 point in the interval (\(\lambda\),f(\(\lambda)\)), then \(x_ 1\) is an unstable fixed point of f. Furthermore in this case, if \(x\in (\lambda,f(\lambda))\), \(x\neq x_ 1\), then the orbit of x leaves this interval in finitely many steps, and there is no periodic point of any period in (\(\lambda\),f(\(\lambda)\)).
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 3953445
zbMATH Keywords: unimodal mapping
zbMATH Keywords: Feigenbaum functional equation
zbMATH Keywords: unimodal solution
zbMATH Keywords: periodic point
reviewed by: Gregory Derfel
MaRDI profile type: Publication
cites work: On Feigenbaum's functional equation gg(lambdax)+lambdag(x)=0
cites work: Universal properties of maps on an interval
cites work: Q3660572
cites work: A computer-assisted proof of the Feigenbaum conjectures
cites work: Stable Orbits and Bifurcation of Maps of the Interval
description: scientific article; zbMATH DE number 3953445