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Fixed points of composition sum operators

Verschueren, Paul and Mestel, Ben D. (2014). Fixed points of composition sum operators. Journal of Difference Equations and Applications, 20(8) pp. 1152–1168.

DOI (Digital Object Identifier) Link: https://doi.org/10.1080/10236198.2014.899357
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Abstract

In the renormalization analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form

fn(z) = Σi=1ai(z)fnii(z)),

where the αi are affine contractions and each ni is either n - 1 or n - 2. We develop a general theory of these fixed points by regarding them as fixed points of ‘composition sum operators’, and apply this theory to test for fixed points in classes of complex analytic functions with various key types of singularities. Finally we demonstrate the construction of the full space of fixed points of one important class, arising from the much studied operator M defined by

Mf(z) = f(-ωz) + f(ω2z + ω), ω = (√5 - 1)/2.

The construction reveals previously unknown solutions.

Item Type: Journal Item
Copyright Holders: 2014 Taylor & Francis
ISSN: 1563-5120
Keywords: renormalization; composition sum operators; fixed point theorems
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 40075
Depositing User: Benjamin Mestel
Date Deposited: 29 Apr 2014 14:18
Last Modified: 10 Dec 2018 09:48
URI: http://oro.open.ac.uk/id/eprint/40075
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