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A generalization of the Babbage functional equation

Year: 2020
Type: article
Abstract: A recent refinement of Kerékjártó's Theorem has shown that in \begin{document}$ \mathbb R $\end{document} and \begin{document}$ \mathbb R^2 $\end{document} all \begin{document}$ \mathcal C^l $\end{document} –solutions of the functional equation \begin{document}$ f^n = \text{Id} $\end{document} are \begin{document}$ \mathcal C^l $\end{document} –linearizable, where \begin{document}$ l\in \{0,1,\dots \infty\} $\end{document} . When \begin{document}$ l\geq 1 $\end{document} , in the real line we prove that the same result holds for solutions of \begin{document}$ f^n = f $\end{document} , while we can only get a local version of it in the plane. Through examples, we show that these results are no longer true when \begin{document}$ l = 0 $\end{document} or when considering the functional equation \begin{document}$ f^n = f^k $\end{document} with \begin{document}$ n>k\geq 2 $\end{document} .
DOI: 10.3934/dcds.2020303
Language: English
Open Access status: diamond
OpenAlex ID: W2999335485
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