A generalisation of the Babbage functional equation

A recent refinement of Kerékjártó's Theorem has shown that in \(\mathbb R\) and \(\mathbb R^2\) all \(\mathcal C^l\)-solutions of the functional equation \(f^n =\textrm{Id}\) are \(\mathcal C^l\)-linearizable, where \(l\in \{0,1,\dots \infty\}\). When \(l\geq 1\), in the real line we prove that...

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Bibliographic Details
Published in:arXiv.org 2020-07
Main Author: Homs-Dones, Marc
Format: Article
Language:English
Subjects:
Functional equations
Online Access:Get full text
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Summary:A recent refinement of Kerékjártó's Theorem has shown that in \(\mathbb R\) and \(\mathbb R^2\) all \(\mathcal C^l\)-solutions of the functional equation \(f^n =\textrm{Id}\) are \(\mathcal C^l\)-linearizable, where \(l\in \{0,1,\dots \infty\}\). When \(l\geq 1\), in the real line we prove that the same result holds for solutions of \(f^n=f\), while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when \(l=0\) or when considering the functional equation \(f^n=f^k\) with \(n>k\geq 2\).
ISSN:2331-8422
DOI:10.48550/arxiv.2001.04573