A generalization of the Babbage functional equation

Marc Homs-Dones

doi:10.3934/dcds.2020303

Article Contents

2021, Volume 41, Issue 2: 899-919. Doi: 10.3934/dcds.2020303

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A generalization of the Babbage functional equation

Marc Homs-Dones

Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom

Received: January 2020

Revised: June 2020

Early access: August 2020

Published: February 2021

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Abstract

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 = \text{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 $.

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Mathematics Subject Classification: Primary: 37C15, 39B12; Secondary: 30D05, 37C05, 37E05, 37E30, 39B22.

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Figure 1. Graph of a generic idempotent continuous function in $ \mathbb R $

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Figure 2. Graph of the function $ f_\lambda $ with $ \lambda = (0.11001\dots)_2 $

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Figure 3. Graph of the function $ f_\lambda $ with $ \lambda = (0.11001\dots)_2 $

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Figure 4. Concatenation of two overhand knots, one figure eight knot and one overhand knot

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