Snapback repellers and semiconjugacy for iterated entire mappings. Global aspects of an old theorem of Poincaré
type: Item
instance of: scholarly article
title: Snapback repellers and semiconjugacy for iterated entire mappings. Global aspects of an old theorem of Poincaré
zbMATH Open document ID: 0760.30009
DOI: 10.1007/BF02567673
published in: Manuscripta Mathematica
publication date: 1992-09-27T00:00:00Z
full work available at URL: https://eudml.org/doc/155698
review text: A theorem of Poincaré guarantees the existence of the local conjugacy of an entire analytic mapping with a hyperbolically unstable fixed point to the horizontal mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. The author especially focuses on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snapback to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. In general there exist infinitely many snapback points and for each of these there exist infinite many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, the author relates the multiplicity of the zeros on the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most a multiplicity of two.
reviewed by: Fred Gross
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F99
zbMATH DE Number: 60074
zbMATH Keywords: critical points
zbMATH Keywords: iterates
author: Franz Rothe
MaRDI profile type: Publication
cites work: Q4177796
cites work: Q4744562
cites work: Complex analytic dynamics on the Riemann sphere
cites work: Chaotic Bursts in Nonlinear Dynamical Systems
cites work: Snap-back repellers imply chaos in \(\mathbb{R}^n\)
cites work: Iteration of analytic functions
cites work: Discrete Dynamical Systems in Dimensions One and Two
OpenAlex ID: W2912193957
description: scientific article; zbMATH DE number 60074