Polynomials without repelling periodic point of given period
type: Item
instance of: scholarly article
title: Polynomials without repelling periodic point of given period
zbMATH Open document ID: 1103.30014
published in: Journal of Mathematical Analysis and Applications
publication date: 2006-12-07T00:00:00Z
review text: Let \(P\) be a polynomial and denote by \(P^k\) the \(k\)-th iterate of \(P.\) A point \(z_0\) is called a periodic point of period \(k\) if \(f^k(z_0)=z_0\) but \(f^j(z_0)\neq z_0\) for \(j<k.\) A periodic point of period \(k\) is called repelling if \(| (f^k)'(z_0)| >1.\) It was shown by the reviewer [Habilitationsschrift, RWTH Aachen (1991)] that if there exists a polynomial of degree \(d\geq 2\) which has no periodic point of period \(k\geq2,\) then \((k,d)\in \{(2,2),(2,3), (2,4),(3,2)\}.\) Moreover, it was shown that exceptional polynomials do indeed occur in each of the four cases. In the present paper all exceptional polynomials are determined; that is, the polynomials for which some iterate fails to have repelling periodic points of some period are classified. It is also shown that the Julia sets of these polynomials are connected.
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 5078042
zbMATH Keywords: periodic point
zbMATH Keywords: fixed point
zbMATH Keywords: period
zbMATH Keywords: multiplier
zbMATH Keywords: repelling
author: Jianming Chang
MaRDI profile type: Publication
full work available at URL: j.jmaa.2005.11.075
OpenAlex ID: W2028204774
cites work: Fixpoints of Polynomials and Rational Functions
cites work: The iteration of cubic polynomials. I: The global topology of parameter space
cites work: Complex dynamics
cites work: Repulsive Fixpoints of Analytic Functions with Applications to Complex Dynamics
DOI: 10.1016/J.JMAA.2005.11.075
description: scientific article; zbMATH DE number 5078042