An iterated implicit complex map
type: Item
instance of: scholarly article
title: An iterated implicit complex map
zbMATH Open document ID: 0604.58025
DOI: 10.1016/0167-2789(86)90025-4
author: Shaun Bullett
author: A. H. Osbaldestin
author: Ian C. Percival
published in: Physica D
publication date: 1986-01-01T00:00:00Z
review text: The authors' ultimate aim is to understand Hamiltonian systems, so they choose to study the iterations of a certain kind of complex mapping that at least possesses an analogue of time-symmetry. After a descriptive motivating survey, they gradually specialise to maps of the form \(z'=f(z)\) that solve an implicit equation \(g(z',z)=0\) where g is quadratic in each of z' and z. When the fixed points are forced to have suitable locations and multiplicities, the coefficients of g depend on just one arbitrary parameter, the rotation number \(\nu\). After this, the resulting map f is restricted to the upper half-plane, and its orbits and related phenomena are then discussed relative to the Julia-Fatou theory of complex maps. The analysis is partly empirical, though some assertions seem to have suppressed proofs - the style of writing does not clearly separate logical deduction from physical induction. The authors demonstrate that they have chosen an interesting type of mapping, but a satisfactory theory remains to be constructed (presumably by studying iterations on the entire complex curve \(g=0)\).
reviewed by: H.B.Griffiths
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37D15
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 3975932
zbMATH Keywords: iterative dynamics
zbMATH Keywords: complex mappings
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: iterations
zbMATH Keywords: time-symmetry
zbMATH Keywords: rotation number
zbMATH Keywords: Julia-Fatou theory
MaRDI profile type: Publication
full work available at URL: 0167-2789(86)90025-4
OpenAlex ID: W2007427113
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description: scientific article; zbMATH DE number 3975932