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 $.

中文翻译：

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Babbage泛函方程的推广

Kerékjártó定理的最新改进表明，在$ \ mathbb R $和$ \ mathbb R ^ 2 $中，所有$ \ mathcal C ^ l $ –功能方程的解$ f ^ n = \ text {Id} $是$ \ mathcal C ^ l $ –linearizable，其中$ l \ in \ {0,1，\ dots \ infty \} $。当$ l \ geq 1 $时，在实线上我们证明了对于$ f ^ n = f $的解而言，同样的结果成立，而我们只能在飞机上得到它的本地版本。通过示例，我们表明当$ l = 0 $或考虑函数方程$ f ^ n = f ^ k $且$ n> k \ geq 2 $时，这些结果不再成立。